1 The data

data <- readRDS(file="../../data/sub_portugal_5provs_4blocks.rds")
  • Height data from a provenance trial (in Portugal) of maritime pine saplings.

  • Randomized block design. Here I selected 5 provenances and 4 blocks.

  • Saplings have different ages: 11, 15, 20 and 27 month old.

table(data$prov,data$block)
##      
##       34 35 36 38
##   LEI 78 71 72 82
##   MIM 44 45 60 60
##   PIE 26 29 34 34
##   SAC 21 20 18 21
##   VAL 37 43 42 42
table(data$prov,as.factor(data$age))
##      
##       11 15 20 27
##   LEI 92 84 65 62
##   MIM 72 59 40 38
##   PIE 36 33 27 27
##   SAC 23 23 17 17
##   VAL 48 44 37 35
randoms <- c("block","prov","clon","tree")
for (i in randoms){data[,i] <- as.factor( data[,i])}
data <- data %>% mutate(age.sc=(age-mean(age))/sd(age))
data %>% ggplot(aes(x=height)) + geom_histogram(color="darkblue", fill="lightblue") + theme_bw()

data %>% ggplot(aes(x=height, color=as.factor(age))) + geom_histogram(fill="white", alpha=0.5, position="identity") + theme_bw()  +
  facet_wrap(~as.factor(age)) + theme(legend.position = "none")

plot_grid(data %>% ggplot(aes(x=age,y=height)) + geom_point(alpha=0.2) + 
      stat_smooth(method = "lm", col = "red") + theme_bw()  +
      theme(axis.title=element_text(size=16)),
data %>% ggplot(aes(x=age,y=height)) + geom_point(alpha=0.2) + 
      stat_smooth(method = "lm", col = "red", formula = y~poly(x,2)) + theme_bw() +
      theme(axis.title=element_text(size=16)))

plot_grid(data %>% ggplot(aes(x=age,y=log(height))) + geom_point(alpha=0.2) + 
      stat_smooth(method = "lm", col = "red") + theme_bw()  +
      theme(axis.title=element_text(size=16)),
data %>% ggplot(aes(x=age,y=log(height))) + geom_point(alpha=0.2) + 
      stat_smooth(method = "lm", col = "red", formula = y~poly(x,2)) + theme_bw() +
      theme(axis.title=element_text(size=16)))

data %>% ggplot(aes(x=height, color=prov)) + geom_histogram(fill="white", alpha=0.5, position="identity") + theme_bw()  +
  facet_wrap(~prov) + theme(legend.position = "none")

data %>% ggplot(aes(x=height, color=block)) + geom_histogram(fill="white", alpha=0.5, position="identity") + theme_bw()  +
  facet_wrap(~block) + theme(legend.position = "none")

2 “Fixed effects” models

Dummy variables for each level = Regularized intercepts, because we use weakly informative priors. But no information shared between intercepts.

P299 in Statistical Rethinking of McElreath.

2.1 Likelihood = Normal distribution

\[\begin{equation} \begin{aligned} h_{i} & \sim \mathcal{N}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,10)\\ \alpha_{PROV} & \sim \mathcal{N}(0,10)\\ \beta_{age} & \sim \mathcal{N}(0,10) \\ \beta_{age2} & \sim \mathcal{N}(0,10)\\ \sigma & \sim \text{HalfCauchy}(0,25) \end{aligned} \end{equation}\]

Comment about the choice of HalfCauchy prior for \(\sigma\)

Prior recommendations

\(\sigma\) is strictly positive, what priors can we use?

  • Very weakly informative prior: \(\sigma \sim \text{HalfCauchy}(0,25)\). From Gelman (2006): 8-schools example (p430). And here.

  • Weakly informative prior:

    • \(\sigma \sim \text{HalfCauchy}(0,1)\) (McElreath, First version) \(\sigma \sim \text{HalfCauchy}(0,5)\) (Betancourt in 8-schools example)

    • \(\sigma \sim \text{exponential}(1)\) (McElreath, Second version) or \(\sigma \sim \text{exponential}(0.1)\)

    • \(\sigma \sim \text{LogNormal}(0,1)\) (McElreath, Second version)

  • More informative prior : \(\sigma \sim \text{HalfNormal}(0,1)\) or \(\sigma \sim \text{Half-t}(3,0,1)\)


data.list <- list(N=length(data$height),              # Number of observations
                  y=data$height,                      # Response variables
                  age=data$age.sc,                    # Tree age
                  nprov=length(unique(data$prov)),    # Number of provenances
                  nblock=length(unique(data$block)),  # Number of blocks
                  prov=as.numeric(data$prov),         # Provenances
                  bloc=as.numeric(data$block))        # Blocks

Model with sigma ~ HalfCauchy(0,1)

mod1_halfcauchy0_1 = stan_model("mod1_halfcauchy0_1.stan") 
fit.mod1_halfcauchy0_1 <- sampling(mod1_halfcauchy0_1, data = data.list, iter = 2000, chains = 2, cores = 2) 
print(fit.mod1_halfcauchy0_1, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_halfcauchy0_1.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                  mean se_mean    sd    10%    50%    90% n_eff Rhat
## beta_age        75.41    0.13  7.81  65.32  75.45  85.38  3597    1
## alpha_prov[1]   82.63    0.16  9.12  71.16  82.58  94.09  3395    1
## alpha_prov[2]   57.36    0.16  9.53  45.34  57.50  69.51  3637    1
## alpha_prov[3]   34.55    0.16  9.30  22.85  34.47  46.52  3579    1
## alpha_prov[4]   26.45    0.15  9.89  13.30  26.51  39.21  4190    1
## alpha_prov[5]   47.94    0.18  9.48  35.74  47.71  60.15  2835    1
## alpha_block[1]  53.88    0.16  9.36  41.99  53.71  65.96  3594    1
## alpha_block[2]  56.60    0.17  9.18  44.76  56.76  68.85  3074    1
## alpha_block[3]  64.83    0.19  9.62  52.74  64.86  77.08  2631    1
## alpha_block[4]  73.33    0.18  9.87  60.35  73.57  85.66  2930    1
## sigma_y        301.04    0.26 10.26 288.03 300.56 314.24  1585    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:53:28 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Model with sigma ~ HalfCauchy(0,25)

mod1_halfcauchy0_25 = stan_model("mod1_halfcauchy0_25.stan")
fit.mod1_halfcauchy0_25 <- sampling(mod1_halfcauchy0_25, data = data.list, iter = 2000, chains = 2, cores = 2)
print(fit.mod1_halfcauchy0_25, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_halfcauchy0_25.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                  mean se_mean    sd    10%    50%    90% n_eff Rhat
## beta_age        75.38    0.15  8.00  65.31  75.26  85.78  2680    1
## alpha_prov[1]   82.43    0.17  9.49  70.27  82.48  94.94  3137    1
## alpha_prov[2]   57.48    0.17  9.41  45.46  57.64  69.35  3038    1
## alpha_prov[3]   34.51    0.16  9.51  22.02  34.74  46.46  3534    1
## alpha_prov[4]   26.27    0.17  9.69  13.63  26.41  38.43  3410    1
## alpha_prov[5]   47.96    0.17  9.80  35.34  47.84  60.23  3438    1
## alpha_block[1]  53.91    0.16  9.45  42.06  53.82  66.07  3326    1
## alpha_block[2]  57.01    0.17  9.30  45.12  56.86  69.30  3110    1
## alpha_block[3]  64.67    0.17  9.66  52.62  64.59  77.22  3055    1
## alpha_block[4]  73.36    0.19  9.79  61.16  73.51  85.77  2657    1
## sigma_y        300.86    0.28 10.70 287.21 300.66 314.25  1423    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:54:12 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Posterior predictive checks

y_rep <- as.matrix(fit.mod1_halfcauchy0_1, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                            legend.title=element_text(size=18),
                                                            axis.text = element_text(size=18),
                                                            legend.position = c(0.8,0.6))

Let’s vectorize this model.

mod1_vectorized = stan_model("mod1_vectorized.stan")
fit.mod1_vectorized <- sampling(mod1_vectorized, data = data.list, iter = 2000, chains = 2, cores = 2)
print(fit.mod1_vectorized, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_vectorized.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                  mean se_mean    sd    10%    50%    90% n_eff Rhat
## beta_age        75.28    0.16  7.40  65.84  75.19  84.81  2234    1
## alpha_prov[1]   82.50    0.19  9.65  69.89  82.44  94.85  2588    1
## alpha_prov[2]   57.51    0.20  9.47  45.21  57.66  69.37  2226    1
## alpha_prov[3]   34.22    0.18  9.68  21.87  34.08  46.78  2814    1
## alpha_prov[4]   26.27    0.17  9.79  13.36  26.44  38.71  3221    1
## alpha_prov[5]   47.81    0.18  9.92  34.79  48.00  60.22  3145    1
## alpha_block[1]  53.38    0.17  9.68  41.23  53.51  65.91  3082    1
## alpha_block[2]  56.49    0.19  9.35  44.29  56.50  68.59  2376    1
## alpha_block[3]  64.46    0.17  9.70  51.97  64.59  76.84  3259    1
## alpha_block[4]  72.87    0.21  9.46  60.61  72.99  84.94  2128    1
## sigma_y        301.49    0.28 10.67 287.97 301.20 315.36  1439    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:54:57 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Another way of vectorizing the model.

mod1_vectorized2 = stan_model("mod1_vectorized2.stan") 
fit.mod1_vectorized2 <- sampling(mod1_vectorized2, data = data.list, iter = 2000, chains = 2, cores = 2) 
print(fit.mod1_vectorized2, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_vectorized2.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                  mean se_mean    sd    10%    50%    90% n_eff Rhat
## beta_age        75.52    0.15  7.80  65.73  75.59  85.36  2817    1
## alpha_prov[1]   82.55    0.18  9.44  69.88  82.63  94.87  2853    1
## alpha_prov[2]   57.43    0.18  9.60  45.16  57.54  69.76  2955    1
## alpha_prov[3]   34.68    0.16  9.34  22.54  34.59  46.55  3463    1
## alpha_prov[4]   26.20    0.14  9.46  13.98  26.35  38.18  4848    1
## alpha_prov[5]   48.20    0.16  9.20  36.33  48.32  60.48  3462    1
## alpha_block[1]  54.02    0.15  9.27  42.55  53.99  65.56  3753    1
## alpha_block[2]  56.90    0.16  9.35  44.88  56.89  68.84  3458    1
## alpha_block[3]  64.67    0.15  9.23  52.72  64.69  76.54  3704    1
## alpha_block[4]  73.15    0.18  9.12  61.89  72.99  84.83  2561    1
## sigma_y        300.90    0.24 10.08 288.50 300.63 314.34  1828    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:55:40 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Let’s compare the speed of these four models.

get_elapsed_time(fit.mod1_halfcauchy0_25)
##          warmup   sample
## chain:1 2.57108 0.347243
## chain:2 2.74658 0.358630
get_elapsed_time(fit.mod1_halfcauchy0_1)
##          warmup   sample
## chain:1 2.43302 0.326342
## chain:2 2.33127 0.331056
get_elapsed_time(fit.mod1_vectorized)
##          warmup   sample
## chain:1 3.03049 0.367305
## chain:2 3.01369 0.403298
get_elapsed_time(fit.mod1_vectorized2)
##          warmup   sample
## chain:1 3.01181 0.282574
## chain:2 2.79074 0.287144

2.2 Likelihood = Log-normal distribution

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,10)\\ \alpha_{PROV} & \sim \mathcal{N}(0,10)\\ \beta_{age} & \sim \mathcal{N}(0,10) \\ \beta_{age2} & \sim \mathcal{N}(0,10)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
mod1_lognormal = stan_model("mod1_lognormal.stan") 
fit.mod1_lognormal <- sampling(mod1_lognormal, data = data.list, iter = 2000, chains = 2, cores = 2)
## Warning: There were 269 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
## http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded
## Warning: Examine the pairs() plot to diagnose sampling problems
print(fit.mod1_lognormal, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_lognormal.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                mean se_mean   sd   10%  50%  90% n_eff Rhat
## beta_age       0.46    0.00 0.02  0.44 0.46 0.48   809 1.00
## alpha_prov[1]  2.81    0.21 3.50 -1.80 2.85 7.12   266 1.01
## alpha_prov[2]  2.75    0.21 3.50 -1.85 2.79 7.05   266 1.01
## alpha_prov[3]  2.65    0.21 3.50 -1.94 2.68 6.95   267 1.01
## alpha_prov[4]  2.76    0.21 3.50 -1.83 2.81 7.07   267 1.01
## alpha_prov[5]  2.77    0.21 3.50 -1.82 2.80 7.05   266 1.01
## alpha_block[1] 3.04    0.21 3.50 -1.27 3.00 7.64   266 1.01
## alpha_block[2] 3.07    0.21 3.49 -1.21 3.04 7.67   266 1.01
## alpha_block[3] 3.09    0.21 3.50 -1.22 3.05 7.68   267 1.01
## alpha_block[4] 3.19    0.21 3.50 -1.09 3.14 7.81   266 1.01
## sigma_y        0.40    0.00 0.01  0.39 0.40 0.41   866 1.01
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:57:09 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod1_lognormal, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ])  + theme_bw() + theme(legend.text=element_text(size=25),
                                                             legend.title=element_text(size=18),
                                                             axis.text = element_text(size=18),
                                                             legend.position = c(0.8,0.6))

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,1)\\ \alpha_{PROV} & \sim \mathcal{N}(0,1)\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
mod1_2_lognormal = stan_model("mod1_2_lognormal.stan")
fit.mod1_2_lognormal <- sampling(mod1_2_lognormal, data = data.list, iter = 2000, chains = 2, cores = 2)
print(fit.mod1_2_lognormal, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_2_lognormal.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                mean se_mean   sd  10%  50%  90% n_eff Rhat
## beta_age       0.46    0.00 0.02 0.44 0.46 0.48  1158    1
## alpha_prov[1]  2.64    0.02 0.32 2.22 2.65 3.05   345    1
## alpha_prov[2]  2.59    0.02 0.32 2.18 2.59 2.99   347    1
## alpha_prov[3]  2.48    0.02 0.32 2.07 2.49 2.89   354    1
## alpha_prov[4]  2.60    0.02 0.32 2.19 2.60 2.99   363    1
## alpha_prov[5]  2.60    0.02 0.32 2.18 2.60 3.00   347    1
## alpha_block[1] 3.20    0.02 0.32 2.80 3.20 3.61   351    1
## alpha_block[2] 3.23    0.02 0.32 2.83 3.23 3.64   354    1
## alpha_block[3] 3.25    0.02 0.32 2.85 3.25 3.67   355    1
## alpha_block[4] 3.35    0.02 0.32 2.94 3.35 3.76   347    1
## sigma_y        0.40    0.00 0.01 0.38 0.40 0.41  1488    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:58:01 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod1_2_lognormal, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ])  + theme_bw() + theme(legend.text=element_text(size=25),
                                                             legend.title=element_text(size=18),
                                                             axis.text = element_text(size=18),
                                                             legend.position = c(0.8,0.6))

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,1)\\ \alpha_{PROV} & \sim \mathcal{N}(0,1)\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \sigma & \sim \text{Exponential}(1) \end{aligned} \end{equation}\]
mod1_3_lognormal = stan_model("mod1_3_lognormal.stan")
fit.mod1_3_lognormal <- sampling(mod1_3_lognormal, data = data.list, iter = 2000, chains = 2, cores = 2)
print(fit.mod1_3_lognormal, pars = c("beta_age","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_3_lognormal.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                mean se_mean   sd  10%  50%  90% n_eff Rhat
## beta_age       0.46    0.00 0.02 0.44 0.46 0.48   865    1
## alpha_prov[1]  2.68    0.02 0.33 2.27 2.69 3.07   362    1
## alpha_prov[2]  2.63    0.02 0.33 2.22 2.64 3.01   368    1
## alpha_prov[3]  2.52    0.02 0.33 2.12 2.54 2.92   362    1
## alpha_prov[4]  2.64    0.02 0.33 2.22 2.65 3.02   363    1
## alpha_prov[5]  2.64    0.02 0.33 2.24 2.65 3.03   366    1
## alpha_block[1] 3.16    0.02 0.33 2.78 3.16 3.57   366    1
## alpha_block[2] 3.19    0.02 0.33 2.81 3.19 3.60   368    1
## alpha_block[3] 3.21    0.02 0.33 2.82 3.20 3.62   364    1
## alpha_block[4] 3.31    0.02 0.33 2.92 3.30 3.72   365    1
## sigma_y        0.40    0.00 0.01 0.38 0.40 0.41  1276    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 16:58:53 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod1_3_lognormal, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ])  + theme_bw() + theme(legend.text=element_text(size=25),
                                                             legend.title=element_text(size=18),
                                                             axis.text = element_text(size=18),
                                                             legend.position = c(0.8,0.6))

2.3 Likelihood = Normal distribution with a log-transformed response variable

\[\begin{equation} \begin{aligned} \text{log}(h_{i}) & \sim \mathcal{N}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,10)\\ \alpha_{PROV} & \sim \mathcal{N}(0,10)\\ \beta_{age} & \sim \mathcal{N}(0,10) \\ \beta_{age2} & \sim \mathcal{N}(0,10)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
data.list_logy <- list(N=length(data$height),              # Number of observations
                  y=log(data$height),                      # Response variables
                  age=data$age.sc,                            # Tree age
                  nprov=length(unique(data$prov)),         # Number of provenances
                  nblock=length(unique(data$block)),       # Number of blocks
                  prov=as.numeric(data$prov),              # Provenances
                  bloc=as.numeric(data$block))             # Blocks
mod1_logy = stan_model("mod1_logy.stan")
fit.mod1log <- sampling(mod1_logy, data = data.list_logy, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
print(fit.mod1log, pars = c("beta_age","beta_age2","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_logy.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                 mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age        0.46    0.00 0.02  0.44  0.46  0.48   750 1.00
## beta_age2      -0.09    0.00 0.02 -0.11 -0.09 -0.07   594 1.01
## alpha_prov[1]   2.54    0.36 3.49 -2.09  2.60  7.05    94 1.03
## alpha_prov[2]   2.49    0.36 3.49 -2.12  2.56  6.99    94 1.03
## alpha_prov[3]   2.38    0.36 3.49 -2.24  2.43  6.88    95 1.03
## alpha_prov[4]   2.50    0.36 3.49 -2.11  2.56  7.01    94 1.03
## alpha_prov[5]   2.50    0.36 3.49 -2.12  2.55  7.00    94 1.03
## alpha_block[1]  3.31    0.36 3.49 -1.18  3.25  7.94    95 1.03
## alpha_block[2]  3.34    0.36 3.49 -1.15  3.30  7.96    95 1.03
## alpha_block[3]  3.35    0.36 3.49 -1.15  3.31  7.98    95 1.03
## alpha_block[4]  3.46    0.36 3.49 -1.06  3.39  8.08    95 1.03
## sigma_y         0.40    0.00 0.01  0.38  0.40  0.41   685 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:00:20 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod1log, pars = "y_rep")
ppc_dens_overlay(y =log(data$height),y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

\[\begin{equation} \begin{aligned} \text{log}(h_{i}) & \sim \mathcal{N}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,1)\\ \alpha_{PROV} & \sim \mathcal{N}(0,1)\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \sigma & \sim \text{Exponential}(1) \end{aligned} \end{equation}\]
mod1_logy2 = stan_model("mod1_logy2.stan")
fit.mod1log2 <- sampling(mod1_logy2, data = data.list_logy, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
print(fit.mod1log2, pars = c("beta_age","beta_age2","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_logy2.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                 mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age        0.46    0.00 0.02  0.44  0.46  0.48  1022    1
## beta_age2      -0.08    0.00 0.02 -0.11 -0.08 -0.06   909    1
## alpha_prov[1]   2.68    0.02 0.33  2.25  2.68  3.11   455    1
## alpha_prov[2]   2.63    0.02 0.33  2.20  2.63  3.05   458    1
## alpha_prov[3]   2.53    0.02 0.33  2.09  2.52  2.94   457    1
## alpha_prov[4]   2.64    0.02 0.33  2.20  2.63  3.06   459    1
## alpha_prov[5]   2.64    0.02 0.33  2.21  2.64  3.06   458    1
## alpha_block[1]  3.16    0.02 0.33  2.74  3.16  3.59   455    1
## alpha_block[2]  3.19    0.02 0.33  2.78  3.19  3.62   457    1
## alpha_block[3]  3.21    0.02 0.33  2.79  3.21  3.64   457    1
## alpha_block[4]  3.31    0.02 0.33  2.89  3.31  3.74   454    1
## sigma_y         0.40    0.00 0.01  0.38  0.40  0.41  1160    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:01:10 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod1log2, pars = "y_rep")
ppc_dens_overlay(y =log(data$height),y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

\[\begin{equation} \begin{aligned} \text{log}(h_{i}) & \sim \mathcal{N}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,1)\\ \alpha_{PROV} & \sim \mathcal{N}(0,1)\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \sigma & \sim \text{HalfStudentT}(4,0,1) \end{aligned} \end{equation}\]
mod1_logy3 = stan_model("mod1_logy3.stan")
fit.mod1log3 <- sampling(mod1_logy3, data = data.list_logy, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
print(fit.mod1log3, pars = c("beta_age","beta_age2","alpha_prov","alpha_block", "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod1_logy3.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                 mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age        0.46    0.00 0.02  0.44  0.46  0.48   818 1.00
## beta_age2      -0.08    0.00 0.02 -0.11 -0.08 -0.06   977 1.00
## alpha_prov[1]   2.63    0.02 0.32  2.23  2.62  3.07   237 1.02
## alpha_prov[2]   2.58    0.02 0.32  2.19  2.56  3.01   237 1.02
## alpha_prov[3]   2.47    0.02 0.32  2.08  2.46  2.90   239 1.02
## alpha_prov[4]   2.58    0.02 0.33  2.18  2.57  3.03   239 1.02
## alpha_prov[5]   2.59    0.02 0.32  2.19  2.57  3.02   241 1.02
## alpha_block[1]  3.21    0.02 0.32  2.77  3.23  3.61   241 1.02
## alpha_block[2]  3.24    0.02 0.32  2.81  3.26  3.64   236 1.02
## alpha_block[3]  3.26    0.02 0.32  2.82  3.27  3.65   240 1.02
## alpha_block[4]  3.36    0.02 0.32  2.92  3.37  3.76   241 1.02
## sigma_y         0.40    0.00 0.01  0.38  0.40  0.41  1229 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:02:00 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3 Multilevel models

Adaptive regularization

Links:

3.1 One varying intercept (provenance)

3.1.1 Centered parameterization

P357 McElreath (first version).

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{PROV[p]}\\ \alpha_{PROV} & \sim \mathcal{N}(\mu_{\alpha_{PROV}},\sigma_{\alpha_{PROV}})\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \mu_{\alpha_{PROV}} & \sim \mathcal{N}(0,1)\\ \sigma_{\alpha_{PROV}} & \sim \text{HalfCauchy}(0,1)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
data.list_mod2_1 <- list(N=length(data$height),          # Number of observations
                  y=data$height,                         # Response variables
                  age=data$age.sc,                       # Tree age
                  nprov=length(unique(data$prov)),       # Number of provenances
                  prov=as.numeric(data$prov))            # Provenances
mod2_1 = stan_model("mod2_1.stan")   
fit.mod2_1 <- sampling(mod2_1, data = data.list_mod2_1, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: There were 14 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: The largest R-hat is 1.13, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_1, pars = c("beta_age","beta_age2","alpha_prov", "sigma_y","mean_alpha_prov","sigma_alpha_prov"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_1.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                   mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age          0.46    0.00 0.02  0.44  0.46  0.49  1087 1.00
## beta_age2        -0.09    0.00 0.02 -0.11 -0.09 -0.07   696 1.00
## alpha_prov[1]     5.90    0.00 0.03  5.86  5.90  5.94   643 1.00
## alpha_prov[2]     5.86    0.00 0.03  5.82  5.86  5.90   923 1.00
## alpha_prov[3]     5.78    0.00 0.04  5.72  5.78  5.83   206 1.02
## alpha_prov[4]     5.86    0.00 0.04  5.80  5.86  5.91  1403 1.00
## alpha_prov[5]     5.86    0.00 0.03  5.82  5.86  5.90  1138 1.00
## sigma_y           0.40    0.00 0.01  0.39  0.40  0.41  1725 1.00
## mean_alpha_prov   5.24    0.65 1.62  2.04  5.84  5.89     6 1.34
## sigma_alpha_prov  0.71    0.68 1.82  0.03  0.07  3.34     7 1.28
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:02:48 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_1, pars = "y_rep")  
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

posterior <- as.array(fit.mod2_1)
np <- nuts_params(fit.mod2_1)
mcmc_trace(posterior, pars = "alpha_prov[3]", np = np) + 
  xlab("Post-warmup iteration")

mcmc_pairs(posterior, np = np, pars = c("mean_alpha_prov","sigma_alpha_prov","alpha_prov[3]","beta_age","beta_age2"), 
           off_diag_args = list(size = 1, alpha = 1/3),np_style = pairs_style_np(div_size=1, div_shape = 19))

Why is this model poorly fitted?

  • Four points (four ages) is not enough to estimate a second degree polynomial? I tried to remove \(\beta_{age2}\) but it hasn’t changed much. This has reduced the number of divergent transitions. (to 11 divergent transitions). But R-hat was still very high. In particular, for \(\mu_{\alpha_{PROV}}\) (R-hat = 1.53 !).

  • Too vague priors?

3.1.1.1 More informative priors

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{PROV[p]}\\ \alpha_{PROV} & \sim \mathcal{N}(\mu_{\alpha_{PROV}},\sigma_{\alpha_{PROV}})\\ \beta_{age} & \sim \text{LogNormal}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \mu_{\alpha_{PROV}} & \sim \text{LogNormal}(0,1)\\ \sigma_{\alpha_{PROV}} & \sim Exponential(1)\\ \sigma & \sim Exponential(1) \end{aligned} \end{equation}\]
mod2_1_otherpriors = stan_model("mod2_1_otherpriors.stan") 
fit.mod2_1_otherpriors <- sampling(mod2_1_otherpriors, data = data.list_mod2_1, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))  
## Warning: There were 37 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_1_otherpriors, pars = c("beta_age","beta_age2","alpha_prov", "sigma_y","mean_alpha_prov","sigma_alpha_prov"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_1_otherpriors.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                   mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age          0.47       0 0.02  0.44  0.47  0.49   720 1.00
## beta_age2        -0.09       0 0.02 -0.11 -0.09 -0.07   767 1.00
## alpha_prov[1]     5.90       0 0.03  5.86  5.90  5.93   827 1.00
## alpha_prov[2]     5.86       0 0.03  5.82  5.86  5.90  1008 1.00
## alpha_prov[3]     5.79       0 0.04  5.73  5.79  5.84   353 1.01
## alpha_prov[4]     5.86       0 0.04  5.81  5.86  5.91  1112 1.00
## alpha_prov[5]     5.86       0 0.03  5.82  5.86  5.91  1078 1.00
## sigma_y           0.40       0 0.01  0.39  0.40  0.41  1402 1.00
## mean_alpha_prov   5.85       0 0.05  5.80  5.85  5.90   550 1.00
## sigma_alpha_prov  0.07       0 0.06  0.03  0.06  0.13   406 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:03:42 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_1_otherpriors, pars = "y_rep")    
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

posterior_cp <- as.array(fit.mod2_1_otherpriors)
np_cp <- nuts_params(fit.mod2_1_otherpriors)  
mcmc_trace(posterior_cp, pars =c( "alpha_prov[3]","mean_alpha_prov"), np = np_cp) + 
  xlab("Post-warmup iteration")

This model is better. But still not ok, any suggestions? Let’s try the non-centered parametrization.

3.1.2 Non-centered parameterization

Links:


From McElreath, P429 (13.4.2.) of Statistical Rethinking (second version)

\[ \alpha \sim \mathcal{N}(\mu,\sigma)\]

is equivalent to

\[\begin{equation} \begin{aligned} \alpha &= \mu + \beta\\ \beta &\sim \mathcal{N}(0,\sigma) \end{aligned} \end{equation}\]

is equivalent to

\[\begin{equation} \begin{aligned} \alpha &= \mu + z\sigma\\ z &\sim \mathcal{N}(0,1) \end{aligned} \end{equation}\]

No parameters are left inside the prior.


From Updating: A Set of Bayesian Notes. Jeffrey B. Arnold. 20 Multilevel Models

These are two ways of writing the same model. However, they change the parameters that the HMC algorithm is actively sampling and thus can have different sampling performance.

However, neither is universally better.

  • If much data, the non-centered parameterization works better
  • If less data, the centered parameterization works better

And there is currently no ex-ante way to know which will work better, and at what amount of “data” that the performance of one or the other is better. However, one other reason to use the centered parameterization (if it is also scaled), is that the Stan HMC implementation tends to be more efficient if all parameters are on the scale.


Non-centered model equation

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \alpha + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + z_{PROV[p]}\sigma_{PROV}\\ \alpha & \sim \mathcal{N}(0,1) \\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ z_{PROV[p]} & \sim \mathcal{N}(0,1)\\ \sigma_{PROV} & \sim \text{HalfCauchy}(0,1)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
mod2_1_nc = stan_model("mod2_1_nc.stan")
fit.mod2_1_nc <- sampling(mod2_1_nc, data = data.list_mod2_1, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: There were 8 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_1_nc, pars=c("beta_age","beta_age2","z_prov","sigma_prov","sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_1_nc.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##             mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age    0.46    0.00 0.02  0.44  0.46  0.49  1245 1.00
## beta_age2  -0.09    0.00 0.02 -0.11 -0.09 -0.07   935 1.00
## z_prov[1]   0.85    0.02 0.64  0.04  0.83  1.66   828 1.00
## z_prov[2]   0.17    0.02 0.63 -0.63  0.18  0.94  1209 1.00
## z_prov[3]  -0.96    0.02 0.73 -1.94 -0.92 -0.05   969 1.01
## z_prov[4]   0.20    0.02 0.69 -0.66  0.19  1.08  1077 1.00
## z_prov[5]   0.23    0.02 0.64 -0.58  0.23  1.03  1129 1.00
## sigma_prov  0.08    0.00 0.05  0.03  0.06  0.14   175 1.01
## sigma_y     0.40    0.00 0.01  0.39  0.40  0.41  1660 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:04:33 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_1_nc, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

I think I’m doing something wrong here… Let’s try with more informative priors.

3.1.2.1 More informative priors

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \alpha + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + z_{PROV[p]}\sigma_{PROV}\\ \alpha & \sim \mathcal{N}(0,1) \\ \beta_{age} & \sim \text{LogNormal}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ z_{PROV[p]} & \sim \mathcal{N}(0,1)\\ \sigma_{PROV} & \sim \text{Exponential}(1)\\ \sigma & \sim \text{Exponential}(1) \end{aligned} \end{equation}\]
mod2_1_nc_otherpriors = stan_model("mod2_1_nc_otherpriors.stan")
fit.mod2_1_nc_otherpriors <- sampling(mod2_1_nc_otherpriors, data = data.list_mod2_1, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: There were 2 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
print(fit.mod2_1_nc_otherpriors, pars=c("beta_age","beta_age2","z_prov","sigma_prov","sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_1_nc_otherpriors.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##             mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age    0.47    0.00 0.02  0.45  0.47  0.49  1053    1
## beta_age2  -0.09    0.00 0.02 -0.11 -0.09 -0.07  1082    1
## z_prov[1]   0.85    0.02 0.62  0.10  0.81  1.65  1136    1
## z_prov[2]   0.15    0.02 0.61 -0.64  0.15  0.89  1036    1
## z_prov[3]  -1.01    0.03 0.71 -1.94 -0.97 -0.14   780    1
## z_prov[4]   0.14    0.02 0.67 -0.72  0.17  0.98  1116    1
## z_prov[5]   0.22    0.02 0.62 -0.52  0.21  0.98  1291    1
## sigma_prov  0.07    0.00 0.05  0.03  0.06  0.13   392    1
## sigma_y     0.40    0.00 0.01  0.39  0.40  0.41  1469    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:05:31 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_1_nc_otherpriors, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

posterior_cp <- as.array(fit.mod2_1_nc_otherpriors)
np_cp <- nuts_params(fit.mod2_1_nc_otherpriors)  
mcmc_trace(posterior_cp, pars =c( "alpha","sigma_prov"), np = np_cp) + 
  xlab("Post-warmup iteration")

mcmc_pairs(posterior_cp, np = np_cp, pars = c("sigma_y","sigma_prov","alpha","beta_age","beta_age2"), 
           off_diag_args = list(size = 1, alpha = 1/3),np_style = pairs_style_np(div_size=3, div_shape = 19))

A lot better!

3.2 Two varying intercepts (prov and block)

3.2.1 Centered parameterization (what we shouldn’t do)

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \alpha_{BLOCK} & \sim \mathcal{N}(\mu_{\alpha_{BLOCK}},\sigma_{\alpha_{BLOCK}})\\ \alpha_{PROV} & \sim \mathcal{N}(\mu_{\alpha_{PROV}},\sigma_{\alpha_{PROV}})\\ \mu_{\alpha_{PROV}} & \sim \mathcal{N}(0,1)\\ \mu_{\alpha_{BLOCK}}& \sim \mathcal{N}(0,1)\\ \sigma_{\alpha_{PROV}} & \sim \text{HalfCauchy}(0,1)\\ \sigma_{\alpha_{BLOCK}} & \sim \text{HalfCauchy}(0,1)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
data.list_mod2_2 <- list(N=length(data$height),          # Number of observations
                  y=data$height,                         # Response variables
                  age=data$age.sc,                       # Tree age
                  nprov=length(unique(data$prov)),       # Number of provenances
                  nblock=length(unique(data$block)),     # Number of blocks
                  prov=as.numeric(data$prov),            # Provenances
                  bloc=as.numeric(data$block))           # Blocks
mod2_2 = stan_model("mod2_2.stan")
fit.mod2_2 <- sampling(mod2_2, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: There were 43 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_2, pars = c("beta_age","beta_age2",
                           "alpha_prov","alpha_block", 
                           "mean_alpha_prov","sigma_alpha_prov",
                           "mean_alpha_block","sigma_alpha_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_2.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                    mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age           0.46    0.00 0.02  0.44  0.46  0.48   847 1.00
## beta_age2         -0.09    0.00 0.02 -0.11 -0.09 -0.07   838 1.00
## alpha_prov[1]      2.92    0.05 0.76  2.01  2.95  3.89   223 1.00
## alpha_prov[2]      2.88    0.05 0.76  1.95  2.89  3.84   222 1.00
## alpha_prov[3]      2.80    0.05 0.76  1.89  2.82  3.77   223 1.00
## alpha_prov[4]      2.88    0.05 0.76  1.97  2.91  3.84   224 1.00
## alpha_prov[5]      2.89    0.05 0.76  1.97  2.90  3.85   224 1.00
## alpha_block[1]     2.92    0.05 0.76  1.97  2.91  3.84   223 1.00
## alpha_block[2]     2.95    0.05 0.76  1.99  2.93  3.87   222 1.00
## alpha_block[3]     2.96    0.05 0.76  2.00  2.95  3.90   222 1.00
## alpha_block[4]     3.05    0.05 0.76  2.09  3.04  3.97   222 1.00
## mean_alpha_prov    2.87    0.05 0.76  1.95  2.88  3.83   225 1.00
## sigma_alpha_prov   0.08    0.00 0.06  0.03  0.07  0.14   460 1.00
## mean_alpha_block   2.85    0.06 0.76  1.91  2.90  3.77   144 1.00
## sigma_alpha_block  0.23    0.06 0.74  0.04  0.09  0.26   135 1.01
## sigma_y            0.40    0.00 0.01  0.38  0.40  0.41  1423 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:06:49 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_2, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6)) 

McElreath: “[…] note that there is only one global mean parameter \(\alpha\), and both of the varying intercept parameters are centered at zero. We can’t identify a separate mean for each varying intercept type, because both intercepts are added to the same linear prediction. So it is conventional to define varying intercepts with a mean of zero, so there’s no risk of accidentally creating hard-to-identify parameters.” “If you do include a mean for each cluster type, it won’t be the end of the world, however.”

3.2.1.1 More informative priors

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\ \mu_{i} & = \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \beta_{age} & \sim \text{LogNormal}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \alpha_{BLOCK} & \sim \mathcal{N}(\mu_{\alpha_{BLOCK}},\sigma_{\alpha_{BLOCK}})\\ \alpha_{PROV} & \sim \mathcal{N}(\mu_{\alpha_{PROV}},\sigma_{\alpha_{PROV}})\\ \mu_{\alpha_{PROV}} & \sim \mathcal{N}(0,1)\\ \mu_{\alpha_{BLOCK}}& \sim \mathcal{N}(0,1)\\ \sigma_{\alpha_{PROV}} & \sim \text{Exponential}(1)\\ \sigma_{\alpha_{BLOCK}} & \sim \text{Exponential}(1)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
mod2_2_otherpriors = stan_model("mod2_2_otherpriors.stan")
fit.mod2_2_otherpriors <- sampling(mod2_2_otherpriors, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: There were 72 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_2_otherpriors, pars = c("beta_age","beta_age2",
                           "alpha_prov","alpha_block", 
                           "mean_alpha_prov","sigma_alpha_prov",
                           "mean_alpha_block","sigma_alpha_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_2_otherpriors.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                    mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age           0.46    0.00 0.02  0.44  0.46  0.48   991 1.00
## beta_age2         -0.09    0.00 0.02 -0.11 -0.09 -0.07  1088 1.00
## alpha_prov[1]      3.04    0.20 1.81  0.60  3.10  5.40    81 1.02
## alpha_prov[2]      2.99    0.20 1.81  0.58  3.05  5.35    81 1.02
## alpha_prov[3]      2.92    0.20 1.81  0.47  2.98  5.29    81 1.02
## alpha_prov[4]      3.00    0.20 1.81  0.56  3.05  5.36    81 1.02
## alpha_prov[5]      3.00    0.20 1.81  0.58  3.07  5.36    81 1.02
## alpha_block[1]     2.81    0.20 1.81  0.44  2.75  5.25    81 1.02
## alpha_block[2]     2.84    0.20 1.81  0.48  2.80  5.27    81 1.02
## alpha_block[3]     2.85    0.20 1.81  0.50  2.79  5.28    81 1.02
## alpha_block[4]     2.94    0.20 1.81  0.58  2.90  5.39    81 1.02
## mean_alpha_prov    2.99    0.20 1.81  0.54  3.05  5.36    81 1.02
## sigma_alpha_prov   0.08    0.00 0.06  0.03  0.06  0.14   586 1.00
## mean_alpha_block   2.86    0.20 1.81  0.50  2.78  5.30    80 1.02
## sigma_alpha_block  0.11    0.01 0.10  0.04  0.08  0.21   353 1.01
## sigma_y            0.40    0.00 0.01  0.38  0.40  0.41   905 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:08:16 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3.2.2 Centered-parameterization (what we should do)

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma^{2}_{i})\\ \mu_{i} & = \alpha + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \beta_{age} & \sim \mathcal{N}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \alpha & \sim \mathcal{N}(0,1)\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,\sigma_{\alpha_{BLOCK}})\\ \alpha_{PROV} & \sim \mathcal{N}(0,\sigma_{\alpha_{PROV}})\\ \sigma_{\alpha_{PROV}} & \sim \text{HalfCauchy}(0,1)\\ \sigma_{\alpha_{BLOCK}} & \sim \text{HalfCauchy}(0,1)\\ \sigma & \sim \text{HalfCauchy}(0,1) \end{aligned} \end{equation}\]
mod2_3 = stan_model("mod2_3.stan")
fit.mod2_3 <- sampling(mod2_3, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
## Warning: There were 2 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 14. See
## http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: The largest R-hat is 1.11, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_3, pars = c("beta_age","beta_age2",
                           "alpha",
                           "alpha_prov","alpha_block", 
                           "sigma_alpha_prov","sigma_alpha_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_3.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                    mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age           0.46    0.00 0.02  0.44  0.46  0.48   874 1.00
## beta_age2         -0.09    0.00 0.02 -0.11 -0.09 -0.07   835 1.00
## alpha              1.85    0.59 1.95 -0.27  1.45  5.76    11 1.17
## alpha_prov[1]      0.05    0.00 0.05  0.00  0.05  0.10   875 1.00
## alpha_prov[2]      0.00    0.00 0.05 -0.04  0.00  0.06  1018 1.00
## alpha_prov[3]     -0.07    0.00 0.05 -0.13 -0.06 -0.01   748 1.00
## alpha_prov[4]      0.01    0.00 0.05 -0.04  0.01  0.07   950 1.00
## alpha_prov[5]      0.01    0.00 0.05 -0.04  0.01  0.06   950 1.00
## alpha_block[1]     3.94    0.59 1.95  0.03  4.36  6.09    11 1.17
## alpha_block[2]     3.97    0.59 1.95  0.05  4.38  6.09    11 1.17
## alpha_block[3]     3.98    0.59 1.95  0.06  4.39  6.11    11 1.17
## alpha_block[4]     4.08    0.59 1.95  0.16  4.51  6.21    11 1.17
## sigma_alpha_prov   0.08    0.00 0.06  0.03  0.06  0.13   560 1.00
## sigma_alpha_block  4.31    0.59 2.80  0.15  4.19  7.56    22 1.08
## sigma_y            0.40    0.00 0.01  0.38  0.40  0.41  1158 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:11:20 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_3, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25),
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6)) 

3.2.2.1 More informative priors

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma^{2}_{i})\\ \mu_{i} & = \alpha + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \alpha_{BLOCK[b]} + \alpha_{PROV[p]}\\ \beta_{age} & \sim \text{LogNormal}(0,1) \\ \beta_{age2} & \sim \mathcal{N}(0,1)\\ \alpha & \sim \text{LogNormal}(0,1)\\ \alpha_{BLOCK} & \sim \mathcal{N}(0,\sigma_{\alpha_{BLOCK}})\\ \alpha_{PROV} & \sim \mathcal{N}(0,\sigma_{\alpha_{PROV}})\\ \sigma_{\alpha_{PROV}} & \sim \text{Exponential}(1)\\ \sigma_{\alpha_{BLOCK}} & \sim \text{Exponential}(1)\\ \sigma & \sim \text{Exponential}(1) \end{aligned} \end{equation}\]
mod2_3_otherpriors = stan_model("mod2_3_otherpriors.stan")
fit.mod2_3_otherpriors <- sampling(mod2_3_otherpriors, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))
print(fit.mod2_3_otherpriors, pars = c("beta_age","beta_age2",
                           "alpha",
                           "alpha_prov","alpha_block", 
                           "sigma_alpha_prov","sigma_alpha_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_3_otherpriors.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                    mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age           0.46       0 0.02  0.44  0.46  0.48  1315 1.00
## beta_age2         -0.09       0 0.02 -0.11 -0.09 -0.07  1164 1.00
## alpha              5.84       0 0.07  5.76  5.85  5.92   360 1.00
## alpha_prov[1]      0.05       0 0.05  0.00  0.04  0.10   394 1.02
## alpha_prov[2]      0.01       0 0.05 -0.04  0.00  0.06   410 1.02
## alpha_prov[3]     -0.07       0 0.05 -0.13 -0.06 -0.01   525 1.01
## alpha_prov[4]      0.01       0 0.05 -0.04  0.01  0.06   483 1.02
## alpha_prov[5]      0.01       0 0.05 -0.04  0.01  0.06   444 1.02
## alpha_block[1]    -0.04       0 0.06 -0.11 -0.04  0.03   309 1.02
## alpha_block[2]    -0.02       0 0.06 -0.08 -0.02  0.05   298 1.02
## alpha_block[3]     0.00       0 0.06 -0.07  0.00  0.07   292 1.02
## alpha_block[4]     0.09       0 0.06  0.02  0.08  0.16   299 1.02
## sigma_alpha_prov   0.07       0 0.06  0.03  0.06  0.13   370 1.01
## sigma_alpha_block  0.10       0 0.08  0.04  0.08  0.19   681 1.00
## sigma_y            0.40       0 0.01  0.39  0.40  0.41  1709 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:12:09 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

\(\alpha\) is very different between the two models!

With more iterations:

fit.mod2_3_otherpriors_iter3000 <- sampling(mod2_3_otherpriors, data = data.list_mod2_2, iter = 3000, chains = 2, cores = 2, control=list(max_treedepth=14))
print(fit.mod2_3_otherpriors_iter3000, pars = c("beta_age","beta_age2",
                           "alpha",
                           "alpha_prov","alpha_block", 
                           "sigma_alpha_prov","sigma_alpha_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_3_otherpriors.
## 2 chains, each with iter=3000; warmup=1500; thin=1; 
## post-warmup draws per chain=1500, total post-warmup draws=3000.
## 
##                    mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age           0.46       0 0.02  0.44  0.46  0.48  2067    1
## beta_age2         -0.09       0 0.02 -0.11 -0.09 -0.07  1968    1
## alpha              5.85       0 0.08  5.77  5.85  5.93   657    1
## alpha_prov[1]      0.05       0 0.04  0.00  0.05  0.10   785    1
## alpha_prov[2]      0.01       0 0.04 -0.05  0.00  0.06   805    1
## alpha_prov[3]     -0.07       0 0.05 -0.13 -0.06 -0.01  1030    1
## alpha_prov[4]      0.01       0 0.05 -0.05  0.01  0.07   886    1
## alpha_prov[5]      0.01       0 0.05 -0.04  0.01  0.07   813    1
## alpha_block[1]    -0.05       0 0.07 -0.12 -0.04  0.02   671    1
## alpha_block[2]    -0.02       0 0.07 -0.09 -0.02  0.04   674    1
## alpha_block[3]    -0.01       0 0.07 -0.08 -0.01  0.06   654    1
## alpha_block[4]     0.08       0 0.07  0.01  0.08  0.15   681    1
## sigma_alpha_prov   0.08       0 0.05  0.03  0.06  0.13  1089    1
## sigma_alpha_block  0.11       0 0.09  0.04  0.08  0.20   989    1
## sigma_y            0.40       0 0.01  0.38  0.40  0.41  2533    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:12:24 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3.2.3 Non-centered parameterization

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma^{2}_{i})\\ \mu_{i} & = \alpha + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + z_{BLOCK[b]}\sigma_{BLOCK} + z_{PROV[p]}\sigma_{PROV}\\ \beta_{age} & \sim \mathcal{N}(0,10) \\ \beta_{age2} & \sim \mathcal{N}(0,10)\\ \alpha & \sim \mathcal{N}(0,10)\\ z_{BLOCK} & \sim \mathcal{N}(0,1)\\ z_{PROV} & \sim \mathcal{N}(0,1)\\ \sigma_{PROV} & \sim HalfCauchy(0,1)\\ \sigma_{BLOCK} & \sim HalfCauchy(0,1)\\ \sigma & \sim HalfCauchy(0,1) \end{aligned} \end{equation}\]
mod2_4 = stan_model("mod2_4.stan")
fit.mod2_4 <- sampling(mod2_4, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))  
## Warning: There were 53 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_4, pars = c("beta_age","beta_age2",
                           "alpha",
                           "z_prov","z_block", 
                           "sigma_prov","sigma_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_4.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##              mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age     0.46    0.00 0.02  0.44  0.46  0.48  1407 1.00
## beta_age2   -0.09    0.00 0.02 -0.11 -0.09 -0.07  1391 1.00
## alpha        5.75    0.02 0.28  5.63  5.82  5.90   131 1.02
## z_prov[1]    0.84    0.02 0.62  0.10  0.82  1.62  1094 1.00
## z_prov[2]    0.12    0.02 0.64 -0.66  0.15  0.92  1066 1.00
## z_prov[3]   -0.98    0.03 0.75 -1.97 -0.95 -0.04   791 1.00
## z_prov[4]    0.21    0.02 0.69 -0.66  0.22  1.07   808 1.00
## z_prov[5]    0.22    0.02 0.62 -0.56  0.23  0.99   941 1.00
## z_block[1]  -0.35    0.03 0.71 -1.26 -0.33  0.57   477 1.00
## z_block[2]  -0.06    0.03 0.63 -0.85 -0.06  0.73   606 1.00
## z_block[3]   0.10    0.02 0.61 -0.67  0.11  0.87   654 1.00
## z_block[4]   1.15    0.02 0.66  0.35  1.11  2.01   888 1.00
## sigma_prov   0.08    0.00 0.09  0.03  0.06  0.15   514 1.01
## sigma_block  0.17    0.03 0.28  0.04  0.09  0.31   119 1.02
## sigma_y      0.40    0.00 0.01  0.39  0.40  0.41  1292 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:15:14 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_4, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25), 
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

posterior_cp <- as.array(fit.mod2_4)
np_cp <- nuts_params(fit.mod2_4)  
mcmc_trace(posterior_cp, pars = c("alpha", "sigma_block", "sigma_prov","z_block[3]"), np = np_cp) + 
  xlab("Post-warmup iteration")

3.2.3.1 More informative priors

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma^{2}_{i})\\ \mu_{i} & = \alpha + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + z_{BLOCK[b]}\sigma_{BLOCK} + z_{PROV[p]}\sigma_{PROV}\\ \beta_{age} & \sim \text{LogNormal}(0,10) \\ \beta_{age2} & \sim \mathcal{N}(0,10)\\ \alpha & \sim \text{LogNormal}(0,10)\\ z_{BLOCK} & \sim \mathcal{N}(0,1)\\ z_{PROV} & \sim \mathcal{N}(0,1)\\ \sigma_{PROV} & \sim \text{Exponential}(0,1)\\ \sigma_{BLOCK} & \sim \text{Exponential}(1)\\ \sigma & \sim \text{Exponential}(0,1) \end{aligned} \end{equation}\]
mod2_4_otherpriors = stan_model("mod2_4_otherpriors.stan")
fit.mod2_4_otherpriors <- sampling(mod2_4_otherpriors, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14))  
## Warning: There were 277 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: The largest R-hat is 1.18, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod2_4_otherpriors, pars = c("beta_age","beta_age2",
                           "alpha",
                           "z_prov","z_block", 
                           "sigma_prov","sigma_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_4_otherpriors.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##              mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age     0.47    0.00 0.02  0.44  0.47  0.49    16 1.12
## beta_age2   -0.09    0.00 0.02 -0.11 -0.09 -0.07    21 1.09
## alpha        5.78    0.05 0.15  5.54  5.83  5.92    10 1.26
## z_prov[1]    0.94    0.21 0.72  0.04  0.86  2.01    12 1.16
## z_prov[2]    0.11    0.05 0.59 -0.67  0.18  0.76   114 1.03
## z_prov[3]   -0.82    0.17 0.83 -1.93 -0.77  0.38    25 1.10
## z_prov[4]    0.24    0.08 0.68 -0.64  0.37  0.89    65 1.05
## z_prov[5]    0.24    0.05 0.58 -0.53  0.32  0.86   138 1.02
## z_block[1]  -0.48    0.13 0.64 -1.32 -0.48  0.29    24 1.09
## z_block[2]  -0.19    0.07 0.59 -0.94 -0.14  0.41    81 1.03
## z_block[3]  -0.05    0.05 0.53 -0.77 -0.01  0.53   129 1.03
## z_block[4]   0.89    0.06 0.71  0.17  0.70  1.90   125 1.02
## sigma_prov   0.12    0.04 0.14  0.03  0.06  0.33    11 1.26
## sigma_block  0.20    0.10 0.23  0.04  0.10  0.69     5 1.49
## sigma_y      0.40    0.00 0.01  0.39  0.40  0.41   358 1.01
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:16:10 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod2_4_otherpriors, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25), 
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

posterior_cp <- as.array(fit.mod2_4_otherpriors)
np_cp <- nuts_params(fit.mod2_4_otherpriors)  
mcmc_trace(posterior_cp, pars = c("alpha", "sigma_block", "sigma_prov","z_block[3]"), np = np_cp) + 
  xlab("Post-warmup iteration")

Let’s increase the target acceptance (adapt_delta=0.99)

McEleath (Second version) : “[…] the target acceptance rate is controlled by the adapt_delta control parameter. The default is 0.95, which means that it aims to attain a 95% acceptance rate. It tries this during the warmup phase, adjusting the step size of each leapfrog step (go back to Chapter 9 if these terms aren’t familiar). When adapt_delta is set high, it results in a smaller step size, which means a more accurate approximation of the curved surface. It also means more computation, which means a slower chain. Increasing adapt_delta can often, but not always, help with divergent transitions.”

fit.mod2_4_otherpriors_adaptdelta <- sampling(mod2_4_otherpriors, data = data.list_mod2_2, iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14,adapt_delta=0.99))
print(fit.mod2_4_otherpriors_adaptdelta, pars = c("beta_age","beta_age2",
                           "alpha",
                           "z_prov","z_block", 
                           "sigma_prov","sigma_block",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod2_4_otherpriors.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##              mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age     0.46    0.00 0.02  0.44  0.46  0.48  1559    1
## beta_age2   -0.09    0.00 0.02 -0.11 -0.09 -0.07  1519    1
## alpha        5.85    0.00 0.09  5.76  5.85  5.94   674    1
## z_prov[1]    0.81    0.02 0.65 -0.01  0.79  1.64   793    1
## z_prov[2]    0.05    0.02 0.63 -0.75  0.04  0.83   855    1
## z_prov[3]   -1.09    0.02 0.73 -2.05 -1.06 -0.15   923    1
## z_prov[4]    0.10    0.02 0.70 -0.75  0.08  0.98  1316    1
## z_prov[5]    0.16    0.02 0.65 -0.66  0.14  1.00  1036    1
## z_block[1]  -0.58    0.02 0.64 -1.44 -0.57  0.22   758    1
## z_block[2]  -0.27    0.02 0.63 -1.06 -0.25  0.51   895    1
## z_block[3]  -0.10    0.02 0.60 -0.84 -0.09  0.64   812    1
## z_block[4]   1.02    0.03 0.73  0.12  0.99  1.98   837    1
## sigma_prov   0.08    0.00 0.07  0.03  0.06  0.13   606    1
## sigma_block  0.11    0.00 0.10  0.04  0.08  0.21   545    1
## sigma_y      0.40    0.00 0.01  0.38  0.40  0.41  1619    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:16:46 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Longer to run, but it’s ok now ! And similar to model mod2_3_otherpriors (model with centered parameterization and more informative priors).

3.3 Varying intercepts and varying slopes

data.list_mod3 <- list(N=length(data$height),          # Number of observations
                  y=data$height,                         # Response variables
                  age=data$age.sc,                       # Tree age
                  nprov=length(unique(data$prov)),       # Number of provenances
                  nblock=length(unique(data$block)),     # Number of blocks
                  prov=as.numeric(data$prov),            # Provenances
                  bloc=as.numeric(data$block))           # Blocks

3.3.1 Centered parameterization

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\[4pt] \mu_{i} & = \alpha + \alpha_{BLOCK[b]} + \alpha_{PROV[p]} + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + \gamma_{PROV[p]}age_{i} \\[4pt] \begin{bmatrix} \alpha_{PROV[p]} \\ \gamma_{PROV[p]} \end{bmatrix} & \sim \text{MVNormal}\left(\begin{bmatrix} 0 \\ 0 \end{bmatrix},\mathbf{S} \right) \\[4pt] \mathbf{S} & = \begin{pmatrix} \sigma_{\alpha_{PROV[p]}} & 0 \\ 0 & \sigma_{\gamma_{PROV[p]}} \end{pmatrix} \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \begin{pmatrix} \sigma_{\alpha_{PROV[p]}} & 0 \\ 0 & \sigma_{\gamma_{PROV[p]}} \end{pmatrix} \\[4pt] \alpha & \sim \text{LogNormal}(0,1)\\[4pt] \beta_{age} & \sim \text{LogNormal}(0,1) \\[4pt] \beta_{age2} & \sim \mathcal{N}(0,1)\\[4pt] \alpha_{BLOCK} & \sim \mathcal{N}(0,\sigma_{\alpha_{BLOCK}})\\[4pt] \sigma_{\alpha_{BLOCK}} & \sim \text{Exponential}(1)\\[4pt] \sigma & \sim \text{Exponential}(1)\\[4pt] \sigma_{\gamma_{PROV[p]}} & \sim \text{Exponential}(1)\\[4pt] \sigma_{\alpha_{PROV}} & \sim \text{Exponential}(1)\\[4pt] \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} & \sim \text{LKJcorr(2)} \end{aligned} \end{equation}\]

3.3.1.1 mod3_1

In this model, I followed the example from here: Stan code of Statistical Rethinking. 13.3 Example: cross-classified chimpanzees with varying slopes.

Even with 99% acceptance rate (adapt_delta=0.99)` and 3000 iterations, the model had some divergent transitions and small sample sizes.

One thing I didn’t understand with this model code is: where are LKJ and \(\sigma_{\alpha_{BLOCK}}\) priors?

mod3_1 = stan_model("mod3_1.stan")
fit.mod3_1 <- sampling(mod3_1, data = data.list_mod3 , iter = 3000, chains = 2, cores = 2, control=list(max_treedepth=14,adapt_delta=0.99))   
## Warning: There were 3 divergent transitions after warmup. Increasing adapt_delta above 0.99 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod3_1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod3_1.
## 2 chains, each with iter=3000; warmup=1500; thin=1; 
## post-warmup draws per chain=1500, total post-warmup draws=3000.
## 
##                  mean se_mean     sd    10%    50%     90% n_eff Rhat
## beta_age         0.47    0.00   0.03   0.44   0.47    0.50   655 1.01
## beta_age2       -0.09    0.00   0.02  -0.11  -0.09   -0.07  1775 1.00
## alpha            5.85    0.00   0.09   5.76   5.85    5.93   364 1.01
## sigma_y          0.40    0.00   0.01   0.38   0.40    0.41  1888 1.00
## sigma_block      0.11    0.00   0.11   0.04   0.08    0.20   623 1.00
## alpha_block[1]  -0.05    0.00   0.07  -0.12  -0.04    0.02   330 1.01
## alpha_block[2]  -0.02    0.00   0.07  -0.09  -0.02    0.04   328 1.01
## alpha_block[3]  -0.01    0.00   0.07  -0.08  -0.01    0.06   346 1.01
## alpha_block[4]   0.08    0.00   0.07   0.01   0.08    0.15   358 1.01
## Rho_prov[1,1]    1.00     NaN   0.00   1.00   1.00    1.00   NaN  NaN
## Rho_prov[1,2]   -0.23    0.02   0.55  -0.90  -0.31    0.60  1098 1.00
## Rho_prov[2,1]   -0.23    0.02   0.55  -0.90  -0.31    0.60  1098 1.00
## Rho_prov[2,2]    1.00    0.00   0.00   1.00   1.00    1.00   572 1.00
## sigma_prov[1]    0.09    0.00   0.07   0.03   0.07    0.15   558 1.00
## sigma_prov[2]    0.04    0.00   0.04   0.01   0.03    0.08   478 1.00
## alpha_prov[1]    0.05    0.00   0.05   0.00   0.05    0.11   561 1.00
## alpha_prov[2]    0.00    0.00   0.05  -0.05   0.00    0.06   578 1.00
## alpha_prov[3]   -0.07    0.00   0.06  -0.14  -0.07   -0.01   602 1.00
## alpha_prov[4]    0.00    0.00   0.06  -0.05   0.00    0.07   642 1.00
## alpha_prov[5]    0.01    0.00   0.06  -0.04   0.01    0.07   622 1.00
## beta_prov[1]    -0.02    0.00   0.03  -0.05  -0.01    0.01   615 1.01
## beta_prov[2]     0.00    0.00   0.03  -0.04   0.00    0.02   817 1.00
## beta_prov[3]     0.01    0.00   0.03  -0.02   0.01    0.05   974 1.00
## beta_prov[4]     0.02    0.00   0.03  -0.01   0.01    0.06   856 1.00
## beta_prov[5]    -0.01    0.00   0.03  -0.05  -0.01    0.02   665 1.01
## v_prov[1,1]      0.05    0.00   0.05   0.00   0.05    0.11   561 1.00
## v_prov[1,2]     -0.02    0.00   0.03  -0.05  -0.01    0.01   615 1.01
## v_prov[2,1]      0.00    0.00   0.05  -0.05   0.00    0.06   578 1.00
## v_prov[2,2]      0.00    0.00   0.03  -0.04   0.00    0.02   817 1.00
## v_prov[3,1]     -0.07    0.00   0.06  -0.14  -0.07   -0.01   602 1.00
## v_prov[3,2]      0.01    0.00   0.03  -0.02   0.01    0.05   974 1.00
## v_prov[4,1]      0.00    0.00   0.06  -0.05   0.00    0.07   642 1.00
## v_prov[4,2]      0.02    0.00   0.03  -0.01   0.01    0.06   856 1.00
## v_prov[5,1]      0.01    0.00   0.06  -0.04   0.01    0.07   622 1.00
## v_prov[5,2]     -0.01    0.00   0.03  -0.05  -0.01    0.02   665 1.01
## SRS_prov[1,1]    0.01    0.00   0.06   0.00   0.00    0.02   639 1.00
## SRS_prov[1,2]    0.00    0.00   0.01   0.00   0.00    0.00  1139 1.00
## SRS_prov[2,1]    0.00    0.00   0.01   0.00   0.00    0.00  1139 1.00
## SRS_prov[2,2]    0.00    0.00   0.01   0.00   0.00    0.01   704 1.00
## y_rep[1]       318.99    2.45 133.12 176.76 292.25  493.19  2961 1.00
## y_rep[2]       219.10    1.63  91.83 121.58 203.01  333.05  3156 1.00
## y_rep[3]       451.93    3.48 188.53 251.21 415.11  700.67  2933 1.00
## y_rep[4]       310.80    2.33 129.14 175.84 287.00  473.67  3064 1.00
## y_rep[5]       216.44    1.65  88.80 122.02 198.29  328.78  2913 1.00
## y_rep[6]       236.48    1.81  97.94 130.81 217.54  365.40  2929 1.00
## y_rep[7]       210.54    1.53  85.14 117.54 196.65  319.30  3096 1.00
## y_rep[8]       313.84    2.45 130.13 176.83 287.54  488.50  2832 1.00
## y_rep[9]       217.75    1.64  92.43 118.95 200.75  337.14  3187 1.00
## y_rep[10]      349.89    2.66 144.03 192.77 327.38  539.58  2935 1.00
## y_rep[11]      452.31    3.30 186.30 250.49 420.76  694.74  3189 1.00
## y_rep[12]      718.14    5.63 307.14 400.93 655.26 1109.42  2978 1.00
## y_rep[13]      324.29    2.42 133.11 181.33 303.50  502.20  3026 1.00
## y_rep[14]      219.23    1.68  89.90 121.55 202.68  337.43  2849 1.00
## y_rep[15]      317.72    2.60 134.06 176.18 293.70  486.94  2650 1.00
## y_rep[16]      520.04    4.08 219.71 285.56 480.24  805.54  2897 1.00
## y_rep[17]      628.25    5.15 267.22 351.33 574.18  983.32  2693 1.00
## y_rep[18]      211.14    1.56  87.32 116.92 195.11  326.18  3152 1.00
## y_rep[19]      236.57    1.93  99.36 130.26 217.46  366.83  2656 1.00
## y_rep[20]      217.01    1.62  90.55 120.61 200.13  332.60  3106 1.00
## y_rep[21]      646.16    4.99 272.18 357.61 600.56  999.58  2980 1.00
## y_rep[22]      359.18    2.68 146.04 198.58 333.78  549.70  2965 1.00
## y_rep[23]      209.11    1.61  88.54 114.33 192.06  330.89  3040 1.00
## y_rep[24]      236.95    1.86 100.29 130.51 217.02  364.48  2900 1.00
## y_rep[25]      325.49    2.37 135.97 181.08 297.77  508.28  3299 1.00
## y_rep[26]      454.59    3.56 197.07 246.73 417.03  713.34  3071 1.00
## y_rep[27]      312.41    2.44 129.23 175.49 287.59  477.27  2799 1.00
## y_rep[28]      476.36    3.83 199.26 264.56 438.67  728.30  2706 1.00
## y_rep[29]      524.10    3.95 220.56 289.01 480.49  815.51  3111 1.00
## y_rep[30]      711.82    5.82 299.89 385.81 651.58 1099.93  2656 1.00
## y_rep[31]      218.52    1.71  90.28 121.97 200.83  339.42  2796 1.00
## y_rep[32]      622.76    4.56 255.41 346.31 573.70  964.86  3139 1.00
## y_rep[33]      215.69    1.65  89.62 119.88 198.42  335.86  2942 1.00
## y_rep[34]      216.64    1.64  88.04 121.56 200.60  330.28  2894 1.00
## y_rep[35]      648.11    5.04 274.42 355.90 596.20 1014.97  2962 1.00
## y_rep[36]      625.61    5.12 259.78 341.91 578.00  968.14  2575 1.00
## y_rep[37]      512.03    4.00 212.99 290.37 469.67  786.98  2831 1.00
## y_rep[38]      312.07    2.32 127.21 176.99 288.02  477.73  3006 1.00
## y_rep[39]      352.24    2.86 147.43 195.15 323.78  542.98  2656 1.00
## y_rep[40]      633.86    4.82 267.32 341.51 587.57  978.47  3074 1.00
## y_rep[41]      469.48    3.52 191.59 264.47 437.47  714.18  2969 1.00
## y_rep[42]      322.33    2.31 130.57 180.54 298.82  494.56  3192 1.00
## y_rep[43]      459.78    3.56 193.10 251.98 420.69  716.99  2940 1.00
## y_rep[44]      209.77    1.53  85.56 117.88 193.45  321.56  3140 1.00
## y_rep[45]      238.07    1.81  98.86 131.70 220.23  368.56  2968 1.00
## y_rep[46]      696.90    5.05 277.67 374.51 654.63 1071.83  3019 1.00
## y_rep[47]      326.31    2.59 136.07 181.44 300.12  511.02  2765 1.00
## y_rep[48]      476.59    3.67 201.23 263.16 441.20  740.67  3001 1.00
## y_rep[49]      320.24    2.48 133.63 175.55 297.85  490.41  2913 1.00
## y_rep[50]      514.22    4.09 219.03 284.12 468.20  800.51  2870 1.00
## y_rep[51]      472.72    3.64 191.98 261.79 439.92  730.84  2789 1.00
## y_rep[52]      632.29    4.86 263.59 342.35 586.56  966.36  2936 1.00
## y_rep[53]      236.61    2.07  99.59 129.97 217.35  368.73  2310 1.00
## y_rep[54]      712.70    5.30 288.05 403.55 661.99 1087.94  2958 1.00
## y_rep[55]      215.46    1.69  90.05 119.87 199.87  331.06  2849 1.00
## y_rep[56]      306.65    2.26 124.20 172.55 285.35  469.40  3008 1.00
## y_rep[57]      215.31    1.61  88.74 118.93 199.44  328.84  3024 1.00
## y_rep[58]      209.24    1.65  90.68 114.60 192.42  321.68  3022 1.00
## y_rep[59]      357.61    2.71 149.54 200.39 327.95  551.35  3049 1.00
## y_rep[60]      657.92    5.36 275.91 364.88 612.42 1008.26  2649 1.00
## y_rep[61]      455.39    3.44 187.65 255.79 420.40  699.29  2969 1.00
## y_rep[62]      321.76    2.55 133.63 177.77 299.92  489.85  2750 1.00
## y_rep[63]      324.47    2.44 133.27 176.73 298.78  504.38  2977 1.00
## y_rep[64]      310.29    2.39 128.54 172.46 287.44  479.37  2893 1.00
## y_rep[65]      211.65    1.60  88.24 117.89 195.19  321.93  3028 1.00
## y_rep[66]      639.23    5.08 273.52 352.30 585.14  996.80  2901 1.00
## y_rep[67]      214.87    1.65  88.77 118.88 199.05  325.46  2884 1.00
## y_rep[68]      515.63    3.79 210.95 285.53 478.77  791.29  3096 1.00
## y_rep[69]      712.03    5.54 296.64 396.10 656.49 1094.66  2867 1.00
## y_rep[70]      466.39    3.55 193.32 257.05 429.92  715.43  2963 1.00
## y_rep[71]      457.64    3.48 193.33 253.94 419.67  703.25  3084 1.00
## y_rep[72]      220.43    1.71  92.85 122.44 203.40  338.82  2962 1.00
## y_rep[73]      235.82    1.71  94.25 132.57 220.64  354.33  3032 1.00
## y_rep[74]      354.50    2.60 145.42 196.05 329.20  544.08  3117 1.00
## y_rep[75]      620.21    4.59 257.26 344.13 574.08  950.34  3141 1.00
## y_rep[76]      322.11    2.35 131.06 180.73 296.79  494.43  3117 1.00
## y_rep[77]      453.07    3.55 190.65 257.57 413.84  701.10  2878 1.00
## y_rep[78]      208.68    1.55  86.89 114.31 193.10  325.13  3125 1.00
## y_rep[79]      236.56    1.78  98.31 133.37 218.65  358.02  3051 1.00
## y_rep[80]      307.41    2.29 126.82 168.53 287.12  472.68  3078 1.00
## y_rep[81]      221.94    1.72  92.98 122.51 204.45  346.75  2929 1.00
## y_rep[82]      215.96    1.63  90.46 120.66 196.57  333.50  3093 1.00
## y_rep[83]      524.20    3.99 216.56 291.43 483.12  815.32  2940 1.00
## y_rep[84]      635.68    4.86 262.28 354.93 588.33  974.36  2917 1.00
## y_rep[85]      321.47    2.41 134.63 180.63 293.96  504.18  3113 1.00
## y_rep[86]      478.19    3.94 202.02 260.97 441.48  739.24  2632 1.00
## y_rep[87]      707.61    5.51 307.52 392.63 648.13 1087.69  3120 1.00
## y_rep[88]      352.00    2.67 142.96 195.17 328.56  537.25  2862 1.00
## y_rep[89]      462.79    3.57 196.97 253.79 427.56  714.09  3050 1.00
## y_rep[90]      621.79    4.50 252.90 343.12 578.36  955.09  3158 1.00
## y_rep[91]      239.39    1.90  99.98 131.62 220.63  368.74  2760 1.00
## y_rep[92]      519.12    3.87 213.53 286.85 482.78  805.03  3039 1.00
## y_rep[93]      213.71    1.62  88.66 120.35 195.82  332.04  2998 1.00
## y_rep[94]      218.23    1.66  92.08 118.01 201.93  337.74  3081 1.00
## y_rep[95]      310.25    2.51 129.67 172.24 285.09  475.32  2664 1.00
## y_rep[96]      323.11    2.55 136.37 177.70 295.42  496.82  2868 1.00
## y_rep[97]      715.07    5.44 301.51 390.89 653.79 1122.67  3073 1.00
## y_rep[98]      354.24    2.63 142.77 198.44 327.47  545.24  2943 1.00
## y_rep[99]      216.79    1.58  88.11 120.03 201.27  326.51  3092 1.00
## y_rep[100]     354.95    2.89 150.13 194.81 327.79  542.28  2697 1.00
## y_rep[101]     628.67    4.67 255.84 347.98 579.27  959.84  3006 1.00
## y_rep[102]     475.13    3.49 194.16 262.76 441.66  728.80  3101 1.00
## y_rep[103]     641.26    4.79 263.34 358.46 590.24  989.08  3019 1.00
## y_rep[104]     326.24    2.59 136.46 180.92 300.16  512.38  2780 1.00
## y_rep[105]     209.78    1.53  82.01 115.83 195.91  319.33  2873 1.00
## y_rep[106]     309.43    2.29 124.72 170.55 287.31  474.94  2969 1.00
## y_rep[107]     724.30    5.44 301.00 401.14 669.09 1121.59  3060 1.00
## y_rep[108]     237.46    1.76  95.31 134.29 218.82  365.90  2931 1.00
## y_rep[109]     320.37    2.40 133.48 177.29 295.94  492.25  3097 1.00
## y_rep[110]     455.96    3.50 190.76 256.09 416.50  708.77  2969 1.00
## y_rep[111]     214.61    1.70  90.07 118.91 199.02  322.86  2809 1.00
## y_rep[112]     653.46    4.88 271.61 362.49 610.11  995.82  3099 1.00
## y_rep[113]     511.72    3.92 218.41 279.64 471.70  787.79  3101 1.00
## y_rep[114]     221.13    1.78  95.24 122.01 202.47  341.66  2849 1.00
## y_rep[115]     463.80    3.52 192.81 257.53 429.57  706.65  2994 1.00
## y_rep[116]     352.11    2.59 141.55 198.41 327.13  539.77  2994 1.00
## y_rep[117]     237.95    1.78  99.74 130.43 219.36  368.50  3123 1.00
## y_rep[118]     321.30    2.46 132.79 176.06 299.09  498.76  2905 1.00
## y_rep[119]     634.95    4.77 256.70 352.15 594.24  965.58  2899 1.00
## y_rep[120]     466.79    3.66 193.56 255.45 435.20  714.93  2802 1.00
## y_rep[121]     516.60    3.98 214.78 287.26 477.24  794.22  2917 1.00
## y_rep[122]     206.53    1.64  87.47 115.10 191.26  316.67  2854 1.00
## y_rep[123]     702.85    5.39 287.79 392.97 651.98 1089.21  2855 1.00
## y_rep[124]     312.32    2.35 128.84 176.10 288.44  480.71  3002 1.00
## y_rep[125]     214.82    1.66  91.88 117.44 197.00  330.99  3071 1.00
## y_rep[126]     219.83    1.67  90.55 122.56 202.39  340.90  2948 1.00
## y_rep[127]     237.13    1.86  99.87 130.23 219.35  363.00  2873 1.00
## y_rep[128]     699.96    5.20 289.58 391.02 650.13 1081.21  3103 1.00
## y_rep[129]     463.74    3.66 196.35 256.56 427.87  710.15  2885 1.00
## y_rep[130]     209.01    1.62  88.30 116.83 193.01  323.72  2972 1.00
## y_rep[131]     322.85    2.54 136.80 178.48 295.56  492.35  2906 1.00
## y_rep[132]     353.50    2.94 148.50 192.86 329.46  545.85  2553 1.00
## y_rep[133]     309.38    2.38 126.89 169.56 289.26  473.34  2842 1.00
## y_rep[134]     326.55    2.62 137.82 179.37 300.53  505.17  2761 1.00
## y_rep[135]     213.99    1.66  86.81 118.72 200.24  326.30  2746 1.00
## y_rep[136]     513.40    3.96 213.63 280.57 471.43  796.12  2903 1.00
## y_rep[137]     651.91    4.86 275.83 362.43 608.27  992.38  3226 1.00
## y_rep[138]     472.46    3.68 195.77 260.42 441.65  723.36  2834 1.00
## y_rep[139]     216.15    1.67  90.66 119.79 200.44  334.26  2955 1.00
## y_rep[140]     635.67    5.32 277.49 353.50 585.70  961.39  2721 1.00
## y_rep[141]     352.40    2.72 143.81 195.58 327.36  532.04  2786 1.00
## y_rep[142]     323.58    2.44 134.68 180.53 297.39  503.19  3041 1.00
## y_rep[143]     642.00    4.91 267.29 350.38 589.58  993.12  2964 1.00
## y_rep[144]     218.37    1.62  88.45 122.89 203.19  333.26  2988 1.00
## y_rep[145]     210.16    1.64  87.16 115.52 193.80  328.81  2817 1.00
## y_rep[146]     318.30    2.45 131.01 179.00 295.76  486.03  2859 1.00
## y_rep[147]     452.97    3.38 186.17 247.74 423.19  697.21  3039 1.00
## y_rep[148]     237.61    1.77  95.54 131.62 221.79  364.30  2924 1.00
## y_rep[149]     706.15    5.63 284.94 395.01 650.28 1075.23  2557 1.00
## y_rep[150]     313.80    2.37 126.89 177.00 290.47  481.97  2856 1.00
## y_rep[151]     463.88    3.84 192.31 257.99 429.38  715.14  2507 1.00
## y_rep[152]     214.97    1.74  88.18 120.67 197.56  334.98  2562 1.00
## y_rep[153]     625.28    4.79 268.66 341.18 574.83  963.38  3150 1.00
## y_rep[154]     522.18    4.02 218.14 287.69 481.66  815.54  2939 1.00
## y_rep[155]     353.03    2.61 144.20 198.14 327.42  543.48  3053 1.00
## y_rep[156]     216.56    1.66  90.16 120.15 200.45  333.50  2954 1.00
## y_rep[157]     649.53    5.12 266.30 361.11 602.13  998.70  2705 1.00
## y_rep[158]     219.47    1.69  91.53 123.57 202.16  334.19  2938 1.00
## y_rep[159]     712.49    5.42 297.54 395.19 662.12 1096.26  3019 1.00
## y_rep[160]     311.41    2.39 129.89 175.81 283.62  480.79  2950 1.00
## y_rep[161]     457.36    3.51 188.56 253.36 420.73  701.69  2888 1.00
## y_rep[162]     466.79    3.41 189.19 260.91 434.68  706.28  3084 1.00
## y_rep[163]     237.45    1.81  99.40 129.75 218.70  363.20  3029 1.00
## y_rep[164]     623.11    4.86 260.25 344.09 574.85  961.38  2864 1.00
## y_rep[165]     519.00    4.03 218.21 291.84 477.32  799.45  2929 1.00
## y_rep[166]     324.04    2.44 133.09 181.28 299.22  501.50  2982 1.00
## y_rep[167]     210.43    1.65  87.68 116.59 193.81  325.03  2815 1.00
## y_rep[168]     319.39    2.45 134.15 176.65 292.45  491.33  2988 1.00
## y_rep[169]     209.32    1.56  86.75 115.43 195.56  320.24  3084 1.00
## y_rep[170]     218.36    1.75  93.15 120.15 198.95  344.28  2842 1.00
## y_rep[171]     641.76    4.96 263.36 358.94 592.44  976.29  2825 1.00
## y_rep[172]     656.20    5.18 275.89 361.49 598.73 1026.20  2840 1.00
## y_rep[173]     215.09    1.75  92.19 117.74 198.38  331.53  2762 1.00
## y_rep[174]     240.58    1.89 102.38 132.09 220.26  374.62  2937 1.00
## y_rep[175]     470.37    3.57 193.57 262.30 435.49  724.12  2941 1.00
## y_rep[176]     325.15    2.43 138.45 177.58 298.85  499.69  3236 1.00
## y_rep[177]     312.61    2.40 128.87 175.07 287.15  476.46  2886 1.00
## y_rep[178]     315.35    2.32 130.15 176.46 291.99  485.05  3143 1.00
## y_rep[179]     469.54    3.53 197.11 259.50 435.49  719.19  3119 1.00
## y_rep[180]     354.84    2.77 149.65 194.76 326.75  541.40  2918 1.00
## y_rep[181]     214.19    1.62  87.20 119.90 198.58  329.54  2888 1.00
## y_rep[182]     647.51    4.84 257.54 364.64 601.43  979.18  2835 1.00
## y_rep[183]     468.38    3.49 194.49 260.39 433.07  728.77  3112 1.00
## y_rep[184]     218.60    1.66  90.07 123.88 200.59  340.92  2951 1.00
## y_rep[185]     237.78    1.78  97.77 134.51 220.66  367.52  3022 1.00
## y_rep[186]     325.49    2.39 133.69 180.57 304.13  495.24  3124 1.00
## y_rep[187]     211.37    1.59  88.39 117.62 196.62  324.71  3094 1.00
## y_rep[188]     356.12    2.62 149.33 194.92 329.97  546.11  3238 1.00
## y_rep[189]     235.17    1.81  95.60 131.05 217.69  366.77  2796 1.00
## y_rep[190]     210.55    1.53  85.43 118.03 195.82  321.62  3101 1.00
## y_rep[191]     654.92    5.17 276.98 358.79 605.33 1016.49  2875 1.00
## y_rep[192]     326.07    2.58 135.56 178.16 303.25  502.58  2758 1.00
## y_rep[193]     314.21    2.44 129.55 175.12 289.51  483.65  2827 1.00
## y_rep[194]     633.44    4.69 257.97 354.28 586.38  978.10  3020 1.00
## y_rep[195]     215.72    1.64  88.50 120.94 198.70  331.50  2930 1.00
## y_rep[196]     317.21    2.40 131.15 178.97 294.09  482.05  2976 1.00
## y_rep[197]     216.14    1.66  91.59 120.62 198.45  335.74  3045 1.00
## y_rep[198]     474.02    3.70 195.48 261.27 439.76  728.66  2788 1.00
## y_rep[199]     707.04    5.23 296.12 387.01 653.06 1098.88  3202 1.00
## y_rep[200]     520.30    4.19 223.71 285.29 477.34  806.91  2846 1.00
## y_rep[201]     472.09    3.56 194.13 263.29 436.26  725.68  2965 1.00
## y_rep[202]     356.27    2.70 149.04 198.08 326.01  559.82  3043 1.00
## y_rep[203]     326.85    2.52 140.01 180.60 302.60  505.36  3087 1.00
## y_rep[204]     457.14    3.45 192.35 252.66 419.57  709.61  3113 1.00
## y_rep[205]     238.87    1.82 100.20 131.49 219.67  369.11  3040 1.00
## y_rep[206]     465.46    3.55 193.94 257.11 428.07  721.11  2989 1.00
## y_rep[207]     354.92    2.91 155.88 194.14 324.09  559.18  2876 1.00
## y_rep[208]     217.87    1.71  89.48 119.48 200.13  335.48  2746 1.00
## y_rep[209]     317.02    2.35 132.77 177.55 291.99  491.27  3202 1.00
## y_rep[210]     641.20    4.67 258.98 359.79 597.57  966.33  3074 1.00
## y_rep[211]     651.18    5.28 271.81 363.38 600.60  993.27  2650 1.00
## y_rep[212]     513.46    3.88 207.22 286.56 477.01  775.02  2857 1.00
## y_rep[213]     218.80    1.66  91.39 120.05 202.30  340.30  3040 1.00
## y_rep[214]     474.51    3.83 194.87 260.80 443.26  723.97  2594 1.00
## y_rep[215]     208.22    1.54  83.66 116.09 192.98  318.30  2943 1.00
## y_rep[216]     315.03    2.66 131.09 175.52 288.92  493.12  2433 1.00
## y_rep[217]     624.48    4.89 267.19 340.14 574.47  978.70  2989 1.00
## y_rep[218]     631.34    4.77 260.84 346.21 582.86  984.17  2994 1.00
## y_rep[219]     208.40    1.61  87.07 113.09 192.84  322.02  2907 1.00
## y_rep[220]     468.74    3.50 193.80 260.31 437.50  714.42  3062 1.00
## y_rep[221]     318.43    2.52 128.63 181.67 294.14  479.03  2608 1.00
## y_rep[222]     219.83    1.66  91.99 122.85 203.45  338.97  3088 1.00
## y_rep[223]     310.71    2.39 129.17 174.20 286.70  472.92  2921 1.00
## y_rep[224]     353.05    2.57 141.67 199.11 330.67  539.87  3035 1.00
## y_rep[225]     455.45    3.60 193.11 252.47 418.41  695.55  2871 1.00
## y_rep[226]     214.39    1.60  88.94 120.82 197.71  330.04  3080 1.00
## y_rep[227]     241.48    1.80  98.89 132.55 224.38  374.76  3029 1.00
## y_rep[228]     469.42    3.59 194.49 259.82 432.54  720.41  2928 1.00
## y_rep[229]     639.16    4.67 262.21 358.13 589.82  978.31  3148 1.00
## y_rep[230]     322.67    2.63 134.80 177.54 299.55  494.78  2634 1.00
## y_rep[231]     655.20    5.19 271.79 365.95 602.02 1012.75  2743 1.00
## y_rep[232]     322.50    2.51 137.43 176.18 296.43  501.85  3003 1.00
## y_rep[233]     516.70    3.88 215.47 280.85 479.11  788.59  3084 1.00
## y_rep[234]     211.08    1.64  86.95 116.09 196.12  326.85  2816 1.00
## y_rep[235]     214.93    1.70  88.30 119.75 199.54  327.59  2695 1.00
## y_rep[236]     468.93    3.68 198.89 263.69 431.19  715.04  2917 1.00
## y_rep[237]     217.30    1.68  90.08 120.66 199.24  330.80  2861 1.00
## y_rep[238]     234.22    1.76  94.80 130.95 217.34  358.49  2886 1.00
## y_rep[239]     311.64    2.32 127.47 173.26 286.91  483.17  3030 1.00
## y_rep[240]     354.25    2.55 142.95 196.33 329.40  546.38  3152 1.00
## y_rep[241]     629.64    4.76 263.11 347.15 580.67  954.02  3057 1.00
## y_rep[242]     453.07    3.32 186.90 250.68 418.36  693.37  3166 1.00
## y_rep[243]     324.62    2.76 134.28 177.73 301.31  500.60  2360 1.00
## y_rep[244]     628.99    4.71 254.20 348.33 583.64  954.28  2908 1.00
## y_rep[245]     712.79    5.35 291.48 397.96 662.15 1077.96  2967 1.00
## y_rep[246]     516.81    4.12 216.58 282.25 479.53  795.31  2767 1.00
## y_rep[247]     318.14    2.44 131.39 180.96 292.55  488.47  2910 1.00
## y_rep[248]     704.39    5.39 287.37 384.10 654.23 1085.11  2844 1.00
## y_rep[249]     213.48    1.59  88.59 117.11 196.69  333.55  3091 1.00
## y_rep[250]     645.79    4.63 263.29 355.75 594.94  989.77  3238 1.00
## y_rep[251]     212.02    1.52  86.01 118.55 196.71  326.06  3199 1.00
## y_rep[252]     324.88    2.50 136.15 182.77 298.21  503.97  2972 1.00
## y_rep[253]     472.69    3.57 197.87 262.08 436.03  732.56  3073 1.00
## y_rep[254]     472.52    3.66 199.82 262.19 434.43  722.71  2976 1.00
## y_rep[255]     314.82    2.42 128.36 173.64 293.02  485.87  2817 1.00
## y_rep[256]     216.00    1.62  89.06 117.89 200.04  333.93  3007 1.00
## y_rep[257]     646.35    4.97 267.33 355.62 598.65  996.91  2890 1.00
## y_rep[258]     633.96    4.92 270.14 351.55 584.38  979.84  3017 1.00
## y_rep[259]     238.26    1.82  99.61 133.83 219.73  366.45  3008 1.00
## y_rep[260]     454.37    3.62 190.13 250.40 423.91  688.14  2751 1.00
## y_rep[261]     351.62    2.63 144.26 194.36 325.60  539.10  3004 1.00
## y_rep[262]     461.34    3.49 188.16 257.30 427.50  706.93  2914 1.00
## y_rep[263]     639.46    4.96 267.19 351.18 588.60 1007.31  2906 1.00
## y_rep[264]     207.56    1.58  85.32 115.40 191.87  316.82  2899 1.00
## y_rep[265]     353.71    2.65 142.67 199.05 325.89  545.67  2888 1.00
## y_rep[266]     214.92    1.72  88.07 119.74 199.87  327.67  2637 1.00
## y_rep[267]     477.44    3.73 202.72 261.35 438.94  743.21  2947 1.00
## y_rep[268]     521.41    4.14 215.67 287.77 483.40  796.13  2719 1.00
## y_rep[269]     315.10    2.30 127.07 177.57 291.43  478.42  3065 1.00
## y_rep[270]     237.29    1.92  98.07 131.29 217.99  364.31  2610 1.00
## y_rep[271]     699.60    5.34 287.87 379.60 650.96 1071.19  2902 1.00
## y_rep[272]     216.93    1.73  88.57 122.45 200.16  333.13  2629 1.00
## y_rep[273]     641.70    5.03 268.21 353.19 595.23  993.72  2840 1.00
## y_rep[274]     319.21    2.39 130.22 178.37 297.01  490.42  2963 1.00
## y_rep[275]     324.24    2.36 133.25 178.54 304.39  499.33  3193 1.00
## y_rep[276]     209.48    1.73  86.05 115.49 193.37  325.00  2486 1.00
## y_rep[277]     352.87    2.79 146.93 193.20 326.26  542.10  2774 1.00
## y_rep[278]     238.64    1.82 101.94 131.15 218.00  372.93  3141 1.00
## y_rep[279]     629.37    4.87 272.02 348.08 576.66  975.73  3122 1.00
## y_rep[280]     217.20    1.64  90.32 121.16 200.52  334.31  3036 1.00
## y_rep[281]     312.66    2.29 128.93 174.51 289.61  479.00  3175 1.00
## y_rep[282]     327.99    2.60 133.48 182.09 305.50  502.06  2629 1.00
## y_rep[283]     322.74    2.43 133.64 181.09 297.43  495.09  3021 1.00
## y_rep[284]     458.41    3.52 192.80 258.09 420.40  698.94  3008 1.00
## y_rep[285]     215.30    1.65  87.67 119.40 198.74  327.36  2810 1.00
## y_rep[286]     712.63    5.95 295.10 394.44 661.31 1090.81  2461 1.00
## y_rep[287]     518.48    4.04 212.51 289.14 475.64  802.55  2766 1.00
## y_rep[288]     453.83    3.60 188.96 253.96 415.57  699.84  2762 1.00
## y_rep[289]     462.72    3.77 193.91 252.56 424.72  721.39  2645 1.00
## y_rep[290]     218.30    1.64  88.24 121.67 203.09  327.88  2886 1.00
## y_rep[291]     469.10    3.70 196.42 254.61 435.82  716.58  2818 1.00
## y_rep[292]     630.70    5.02 266.35 350.13 581.81  977.10  2810 1.00
## y_rep[293]     643.53    4.92 265.54 350.05 594.86  994.49  2909 1.00
## y_rep[294]     210.26    1.65  86.57 116.66 194.10  326.59  2753 1.00
## y_rep[295]     318.29    2.36 129.28 179.76 295.20  484.48  2996 1.00
## y_rep[296]     314.43    2.49 129.07 174.69 292.22  478.49  2679 1.00
## y_rep[297]     216.99    1.70  90.85 120.02 199.61  335.06  2870 1.00
## y_rep[298]     518.71    4.16 220.74 286.36 478.47  797.28  2810 1.00
## y_rep[299]     653.64    5.89 279.82 355.45 598.15 1021.03  2258 1.00
## y_rep[300]     323.04    2.53 135.84 175.89 296.88  496.35  2892 1.00
## y_rep[301]     710.33    5.65 304.46 382.56 656.24 1096.85  2908 1.00
## y_rep[302]     351.59    2.66 145.74 198.11 323.54  534.98  3004 1.00
## y_rep[303]     237.51    1.82  98.06 130.60 220.22  366.95  2916 1.00
## y_rep[304]     203.10    1.53  82.48 113.72 188.34  312.95  2922 1.00
## y_rep[305]     304.66    2.29 127.20 169.24 281.19  465.75  3089 1.00
## y_rep[306]     207.79    1.56  88.16 114.25 192.01  321.31  3191 1.00
## y_rep[307]     297.79    2.30 124.24 164.53 271.48  462.97  2907 1.00
## y_rep[308]     226.67    1.72  94.27 124.31 210.55  352.65  2989 1.00
## y_rep[309]     311.19    2.44 130.90 172.76 284.77  486.35  2872 1.00
## y_rep[310]     453.04    3.31 183.32 255.87 418.69  692.45  3059 1.00
## y_rep[311]     198.26    1.50  81.14 109.87 185.78  300.56  2908 1.00
## y_rep[312]     312.89    2.30 129.53 173.44 290.51  481.08  3166 1.00
## y_rep[313]     226.14    1.76  95.10 121.71 209.72  349.05  2916 1.00
## y_rep[314]     456.30    3.38 183.24 257.55 424.34  694.07  2947 1.00
## y_rep[315]     207.02    1.64  89.26 115.85 188.85  321.34  2946 1.00
## y_rep[316]     196.30    1.48  80.41 109.76 182.11  298.11  2955 1.00
## y_rep[317]     436.49    3.31 182.76 240.40 402.54  669.82  3048 1.00
## y_rep[318]     203.45    1.52  83.36 115.64 189.43  307.91  3005 1.00
## y_rep[319]     307.50    2.47 126.23 166.90 284.88  472.30  2621 1.00
## y_rep[320]     295.78    2.29 122.16 160.97 274.43  454.65  2834 1.00
## y_rep[321]     632.53    5.11 261.33 352.01 588.01  961.38  2617 1.00
## y_rep[322]     609.32    4.95 255.37 331.88 566.76  942.57  2665 1.00
## y_rep[323]     202.28    1.55  85.55 111.25 186.22  308.88  3058 1.00
## y_rep[324]     694.52    5.46 299.36 375.56 640.22 1082.80  3001 1.00
## y_rep[325]     204.76    1.48  81.07 115.31 191.15  315.50  2989 1.00
## y_rep[326]     227.63    1.83  96.66 124.38 210.22  352.12  2802 1.00
## y_rep[327]     295.40    2.24 121.35 165.15 271.59  451.90  2942 1.00
## y_rep[328]     310.11    2.32 126.60 174.63 286.67  466.44  2978 1.00
## y_rep[329]     304.21    2.37 123.16 170.10 282.41  464.54  2696 1.00
## y_rep[330]     447.34    3.46 185.39 253.27 411.40  690.59  2864 1.00
## y_rep[331]     621.57    4.65 255.26 351.98 573.87  946.00  3007 1.00
## y_rep[332]     200.49    1.57  86.33 111.25 184.51  306.35  3021 1.00
## y_rep[333]     339.83    2.68 144.14 188.11 312.56  519.27  2893 1.00
## y_rep[334]     497.11    3.73 199.43 278.71 465.48  764.55  2852 1.00
## y_rep[335]     501.25    3.87 210.69 275.60 465.19  770.42  2971 1.00
## y_rep[336]     638.55    4.92 264.32 353.97 591.50  980.38  2892 1.00
## y_rep[337]     695.32    5.65 290.68 382.78 642.22 1060.80  2651 1.00
## y_rep[338]     207.49    1.63  84.77 115.09 191.63  319.00  2716 1.00
## y_rep[339]     308.50    2.20 124.66 173.37 287.81  469.00  3220 1.00
## y_rep[340]     198.32    1.49  82.59 110.54 183.69  308.22  3093 1.00
## y_rep[341]     204.23    1.58  83.26 114.41 189.36  312.59  2787 1.00
## y_rep[342]     224.66    1.66  93.16 123.73 207.63  347.56  3144 1.00
## y_rep[343]     336.71    2.52 140.31 187.46 311.12  516.45  3088 1.00
## y_rep[344]     454.27    3.24 181.37 255.33 421.72  688.02  3134 1.00
## y_rep[345]     200.27    1.51  82.01 112.06 185.94  303.64  2942 1.00
## y_rep[346]     308.39    2.46 136.79 168.96 283.24  478.29  3083 1.00
## y_rep[347]     205.68    1.55  83.89 114.98 192.52  314.66  2923 1.00
## y_rep[348]     311.64    2.46 132.23 172.21 286.16  481.94  2880 1.00
## y_rep[349]     629.45    5.08 264.16 350.20 579.99  991.77  2705 1.00
## y_rep[350]     296.86    2.21 123.33 164.06 273.81  457.24  3126 1.00
## y_rep[351]     448.43    3.61 191.63 251.65 413.81  686.21  2824 1.00
## y_rep[352]     496.89    3.72 205.44 274.68 459.05  757.91  3055 1.00
## y_rep[353]     332.21    2.60 137.94 182.31 306.34  516.83  2823 1.00
## y_rep[354]     202.46    1.56  82.52 112.05 188.43  309.55  2791 1.00
## y_rep[355]     700.85    5.54 302.67 387.09 641.04 1087.52  2982 1.00
## y_rep[356]     633.15    4.68 262.62 352.45 584.39  981.26  3152 1.00
## y_rep[357]     224.39    1.64  90.58 126.14 208.12  341.97  3040 1.00
## y_rep[358]     459.36    3.40 189.45 253.75 425.61  704.03  3114 1.00
## y_rep[359]     639.97    4.78 262.76 360.72 590.59  983.02  3017 1.00
## y_rep[360]     447.91    3.76 189.12 246.58 411.78  688.75  2536 1.00
## y_rep[361]     226.66    1.84  93.67 124.20 210.41  353.36  2580 1.00
## y_rep[362]     499.35    3.92 207.93 280.31 458.68  765.97  2815 1.00
## y_rep[363]     309.23    2.43 131.30 169.92 285.28  475.74  2909 1.00
## y_rep[364]     204.36    1.54  83.34 116.48 188.49  313.08  2911 1.00
## y_rep[365]     297.77    2.24 125.40 160.05 276.70  463.75  3129 1.00
## y_rep[366]     202.15    1.52  84.99 110.72 186.75  313.78  3133 1.00
## y_rep[367]     699.81    5.23 288.18 389.79 646.55 1077.87  3031 1.00
## y_rep[368]     632.39    4.81 260.99 354.64 587.50  973.53  2947 1.00
## y_rep[369]     199.62    1.45  80.59 109.97 186.58  304.59  3091 1.00
## y_rep[370]     341.04    2.67 143.19 191.69 311.86  526.51  2881 1.00
## y_rep[371]     456.60    3.53 191.59 252.95 422.94  690.65  2949 1.00
## y_rep[372]     305.52    2.22 122.54 170.04 283.37  467.87  3043 1.00
## y_rep[373]     202.07    1.54  82.38 113.92 186.20  312.04  2873 1.00
## y_rep[374]     625.43    4.71 262.32 345.95 575.37  971.27  3108 1.00
## y_rep[375]     605.99    5.43 252.99 336.38 555.01  938.67  2173 1.00
## y_rep[376]     203.06    1.50  86.86 112.67 184.85  313.06  3357 1.00
## y_rep[377]     692.11    5.04 276.07 393.17 638.23 1057.09  3004 1.00
## y_rep[378]     199.37    1.53  82.14 109.86 183.97  310.11  2866 1.00
## y_rep[379]     452.17    3.40 186.01 252.91 417.75  697.10  2990 1.00
## y_rep[380]     297.60    2.27 126.96 163.53 272.57  466.29  3140 1.00
## y_rep[381]     309.29    2.35 128.85 170.94 286.17  474.69  3015 1.00
## y_rep[382]     502.23    3.88 210.39 276.70 462.68  776.21  2934 1.00
## y_rep[383]     441.97    3.18 180.75 243.53 410.71  671.27  3222 1.00
## y_rep[384]     335.49    2.55 139.82 185.40 305.79  524.69  2996 1.00
## y_rep[385]     224.06    1.76  92.10 125.04 208.08  346.99  2732 1.00
## y_rep[386]     300.25    2.28 125.62 162.68 279.10  463.27  3028 1.00
## y_rep[387]     295.51    2.22 121.70 163.26 272.49  455.97  2996 1.00
## y_rep[388]     227.69    1.76  95.31 125.57 210.15  352.76  2922 1.00
## y_rep[389]     602.71    4.35 238.55 342.09 562.24  920.35  3011 1.00
## y_rep[390]     634.12    4.59 265.17 353.80 583.52  974.71  3331 1.00
## y_rep[391]     499.89    3.67 201.40 282.20 461.40  761.44  3008 1.00
## y_rep[392]     308.80    2.39 128.08 171.01 285.70  480.57  2879 1.00
## y_rep[393]     452.43    3.31 179.98 252.78 423.41  694.04  2961 1.00
## y_rep[394]     439.14    3.33 182.09 244.05 407.13  674.66  2988 1.00
## y_rep[395]     204.44    1.56  84.65 113.87 187.54  319.94  2938 1.00
## y_rep[396]     198.26    1.61  83.53 108.79 183.17  303.54  2697 1.00
## y_rep[397]     206.15    1.60  86.59 113.36 191.20  317.36  2944 1.00
## y_rep[398]     336.90    2.45 136.84 187.40 315.67  513.05  3116 1.00
## y_rep[399]     225.41    1.75  95.57 123.27 208.05  345.24  2984 1.00
## y_rep[400]     196.73    1.43  79.56 109.31 183.55  302.24  3101 1.00
## y_rep[401]     686.13    5.49 285.17 379.77 623.90 1064.71  2696 1.00
## y_rep[402]     199.44    1.47  80.69 112.03 185.15  303.37  3017 1.00
## y_rep[403]     504.46    3.74 210.12 279.62 465.35  766.02  3161 1.00
## y_rep[404]     206.17    1.60  87.57 113.51 189.52  318.43  3012 1.00
## y_rep[405]     442.67    3.57 188.80 245.15 409.98  675.94  2799 1.00
## y_rep[406]     605.28    4.47 253.74 338.74 552.10  950.24  3220 1.00
## y_rep[407]     334.55    2.73 141.52 182.93 307.99  515.71  2693 1.00
## y_rep[408]     294.90    2.20 119.31 166.58 275.05  446.68  2933 1.00
## y_rep[409]     302.27    2.21 121.71 171.60 280.09  465.75  3020 1.00
## y_rep[410]     308.21    2.45 132.92 169.76 281.97  474.87  2938 1.00
## y_rep[411]     197.17    1.56  82.88 110.35 180.85  304.87  2838 1.00
## y_rep[412]     630.89    4.76 263.55 349.41 583.87  975.59  3072 1.00
## y_rep[413]     303.28    2.21 122.64 171.54 279.92  466.18  3076 1.00
## y_rep[414]     450.16    3.49 193.69 241.12 417.61  692.13  3083 1.00
## y_rep[415]     223.96    1.67  92.94 123.76 207.12  345.87  3083 1.00
## y_rep[416]     336.49    2.43 137.85 189.70 309.70  512.84  3223 1.00
## y_rep[417]     297.44    2.28 123.42 162.74 273.61  459.73  2928 1.00
## y_rep[418]     627.85    4.79 258.92 347.95 578.54  965.38  2919 1.00
## y_rep[419]     203.15    1.59  86.00 111.02 187.86  309.42  2924 1.00
## y_rep[420]     462.27    3.50 189.90 250.91 429.84  707.29  2949 1.00
## y_rep[421]     205.58    1.57  85.34 114.35 189.17  316.22  2937 1.00
## y_rep[422]     333.95    2.43 135.48 187.40 307.91  509.36  3112 1.00
## y_rep[423]     451.84    3.46 191.19 250.39 415.24  698.30  3048 1.00
## y_rep[424]     306.68    2.37 129.47 168.26 281.00  473.62  2993 1.00
## y_rep[425]     301.00    2.32 127.11 162.57 278.65  466.18  2999 1.00
## y_rep[426]     205.22    1.56  85.15 114.87 189.35  314.94  2963 1.00
## y_rep[427]     491.07    3.75 201.47 273.33 456.03  746.14  2884 1.00
## y_rep[428]     304.02    2.17 122.05 167.65 279.22  469.71  3161 1.00
## y_rep[429]     225.38    1.80  94.63 123.86 208.57  350.37  2752 1.00
## y_rep[430]     690.39    5.34 283.84 384.20 633.93 1056.23  2829 1.00
## y_rep[431]     197.66    1.58  80.08 110.11 184.19  304.03  2566 1.00
## y_rep[432]     455.39    3.44 191.33 247.06 418.49  703.09  3095 1.00
## y_rep[433]     625.49    5.04 269.31 336.78 577.11  964.94  2860 1.00
## y_rep[434]     202.65    1.60  86.67 112.38 185.58  314.81  2950 1.00
## y_rep[435]     633.07    4.66 258.92 345.53 586.20  972.14  3086 1.00
## y_rep[436]     296.57    2.46 122.55 163.89 274.08  455.48  2488 1.00
## y_rep[437]     496.93    3.79 201.73 274.21 460.66  770.39  2827 1.00
## y_rep[438]     198.45    1.64  83.04 109.36 182.78  308.01  2558 1.00
## y_rep[439]     222.47    1.62  90.24 125.22 205.88  339.71  3111 1.00
## y_rep[440]     205.48    1.59  86.70 114.19 190.05  313.13  2982 1.00
## y_rep[441]     700.17    5.55 295.14 382.37 645.66 1086.64  2826 1.00
## y_rep[442]     337.46    2.64 142.58 188.01 309.16  520.52  2907 1.00
## y_rep[443]     204.97    1.55  85.11 114.26 188.92  318.15  3013 1.00
## y_rep[444]     225.31    1.74  96.59 122.98 207.24  346.69  3068 1.00
## y_rep[445]     612.81    4.70 252.87 339.45 563.99  937.22  2889 1.00
## y_rep[446]     639.04    4.69 261.04 358.15 594.80  971.00  3093 1.00
## y_rep[447]     437.11    3.36 185.97 239.52 399.71  672.30  3072 1.00
## y_rep[448]     694.94    5.49 295.24 383.75 637.72 1082.04  2888 1.00
## y_rep[449]     502.09    3.92 218.52 275.52 460.39  779.91  3101 1.00
## y_rep[450]     293.60    2.25 120.78 162.51 272.23  450.87  2894 1.00
## y_rep[451]     307.75    2.27 123.61 174.93 286.22  468.55  2975 1.00
## y_rep[452]     454.64    3.61 189.24 250.00 422.35  690.36  2746 1.00
## y_rep[453]     208.35    1.55  86.34 115.66 192.72  318.20  3123 1.00
## y_rep[454]     304.05    2.24 123.25 170.32 281.46  470.71  3020 1.00
## y_rep[455]     201.68    1.56  83.52 113.05 186.81  305.22  2851 1.00
## y_rep[456]     199.12    1.52  83.44 109.65 183.40  304.25  3012 1.00
## y_rep[457]     337.38    2.49 138.71 184.20 314.29  517.65  3111 1.00
## y_rep[458]     454.39    3.43 188.59 255.22 420.45  696.81  3021 1.00
## y_rep[459]     197.77    1.53  83.04 107.75 184.24  299.05  2946 1.00
## y_rep[460]     204.99    1.50  82.91 115.97 189.64  310.99  3050 1.00
## y_rep[461]     301.57    2.46 125.98 163.22 280.97  465.10  2615 1.00
## y_rep[462]     224.43    1.73  93.31 123.45 208.02  347.42  2901 1.00
## y_rep[463]     204.25    1.62  84.16 114.36 188.01  317.67  2687 1.00
## y_rep[464]     306.24    2.39 127.52 169.50 283.58  467.19  2849 1.00
## y_rep[465]     201.72    1.51  82.17 111.93 185.89  313.58  2946 1.00
## y_rep[466]     198.99    1.57  84.61 108.70 183.89  306.50  2920 1.00
## y_rep[467]     223.59    1.67  90.86 125.51 207.14  344.49  2966 1.00
## y_rep[468]     491.49    3.75 202.99 276.97 453.94  747.10  2934 1.00
## y_rep[469]     307.52    2.30 126.21 172.62 285.30  468.76  3013 1.00
## y_rep[470]     299.37    2.27 123.35 165.30 277.42  459.39  2944 1.00
## y_rep[471]     334.52    2.53 138.51 183.30 309.32  514.15  2993 1.00
## y_rep[472]     697.09    5.24 288.10 392.29 642.64 1071.21  3020 1.00
## y_rep[473]     644.30    5.02 270.27 356.83 592.25  995.33  2897 1.00
## y_rep[474]     451.92    3.42 190.32 251.01 418.94  688.21  3092 1.00
## y_rep[475]     205.58    1.60  86.15 114.79 188.04  319.88  2903 1.00
## y_rep[476]     453.44    3.36 187.58 252.79 421.84  698.08  3119 1.00
## y_rep[477]     629.11    4.94 266.46 351.14 581.08  969.61  2907 1.00
## y_rep[478]     688.86    5.34 284.45 378.60 637.74 1052.96  2833 1.00
## y_rep[479]     203.90    1.86  87.05 112.28 185.81  321.60  2202 1.00
## y_rep[480]     334.34    2.59 141.21 183.33 307.69  513.60  2967 1.00
## y_rep[481]     228.40    1.72  94.13 126.06 210.28  350.02  2981 1.00
## y_rep[482]     203.59    1.57  85.83 111.65 187.35  320.31  2970 1.00
## y_rep[483]     199.19    1.52  84.18 110.08 182.02  310.11  3076 1.00
## y_rep[484]     311.01    2.32 128.74 170.34 285.82  487.50  3091 1.00
## y_rep[485]     501.61    4.01 210.26 277.43 461.14  774.08  2746 1.00
## y_rep[486]     302.60    2.27 123.58 167.07 281.73  464.62  2970 1.00
## y_rep[487]     619.94    5.15 258.38 343.75 573.47  950.87  2514 1.00
## y_rep[488]     224.84    1.72  91.90 123.05 209.65  346.15  2859 1.00
## y_rep[489]     448.85    3.23 184.17 250.61 411.84  685.70  3245 1.00
## y_rep[490]     334.38    2.63 137.08 185.30 308.82  515.88  2716 1.00
## y_rep[491]     446.39    3.21 177.91 248.19 416.50  679.05  3078 1.00
## y_rep[492]     504.80    4.22 217.17 278.13 463.58  783.23  2649 1.00
## y_rep[493]     203.65    1.54  84.46 114.18 187.83  313.05  3001 1.00
## y_rep[494]     196.50    1.58  81.95 109.88 179.50  301.88  2700 1.00
## y_rep[495]     294.22    2.18 121.04 162.81 271.04  453.52  3072 1.00
## y_rep[496]     690.99    5.15 279.51 383.08 639.40 1068.46  2949 1.00
## y_rep[497]     204.41    1.58  85.42 112.78 188.94  316.81  2929 1.00
## y_rep[498]     633.51    4.95 265.22 352.55 584.37  969.73  2872 1.00
## y_rep[499]     306.09    2.35 124.90 172.13 287.53  468.55  2819 1.00
## y_rep[500]     298.60    2.23 123.55 163.62 274.46  460.70  3070 1.00
## y_rep[501]     205.57    1.44  84.18 112.21 191.01  316.77  3409 1.00
## y_rep[502]     687.51    5.28 288.32 380.31 625.73 1046.35  2984 1.00
## y_rep[503]     491.29    3.69 204.97 273.10 455.58  764.07  3089 1.00
## y_rep[504]     451.22    3.39 183.16 247.63 422.74  684.06  2916 1.00
## y_rep[505]     635.62    4.77 264.81 350.75 584.98  979.50  3084 1.00
## y_rep[506]     199.41    1.48  81.96 112.06 184.10  306.55  3081 1.00
## y_rep[507]     334.10    2.54 136.88 183.24 310.75  506.66  2906 1.00
## y_rep[508]     226.62    1.74  93.74 124.99 210.20  347.49  2916 1.00
## y_rep[509]     308.55    2.28 124.88 171.88 285.23  476.92  3001 1.00
## y_rep[510]     302.24    2.23 122.74 169.74 280.40  459.38  3034 1.00
## y_rep[511]     197.69    1.52  82.61 108.83 182.67  304.30  2941 1.00
## y_rep[512]     630.68    4.83 262.14 352.92 583.01  969.91  2944 1.00
## y_rep[513]     206.10    1.64  84.46 116.06 189.39  314.63  2654 1.00
## y_rep[514]     314.57    2.58 134.36 170.69 287.65  491.76  2711 1.00
## y_rep[515]     282.43    2.15 117.62 155.86 261.52  439.49  2993 1.00
## y_rep[516]     591.66    4.53 244.44 324.26 547.01  915.33  2915 1.00
## y_rep[517]     181.78    1.37  76.38  99.10 167.90  280.72  3099 1.00
## y_rep[518]     428.48    3.30 175.77 237.96 396.99  657.87  2843 1.00
## y_rep[519]     186.74    1.37  76.41 103.07 173.38  286.77  3132 1.00
## y_rep[520]     420.72    3.33 176.84 227.87 389.51  653.24  2813 1.00
## y_rep[521]     187.32    1.39  76.71 102.79 172.32  287.84  3035 1.00
## y_rep[522]     608.01    4.57 250.73 333.00 568.01  934.85  3011 1.00
## y_rep[523]     280.53    2.24 120.14 154.12 254.77  442.28  2880 1.00
## y_rep[524]     664.97    5.25 288.77 358.13 609.78 1042.69  3020 1.00
## y_rep[525]     470.13    3.64 197.83 263.97 431.54  726.27  2956 1.00
## y_rep[526]     589.84    4.67 250.07 329.15 547.08  908.38  2862 1.00
## y_rep[527]     462.33    3.47 192.74 254.10 427.44  709.39  3085 1.00
## y_rep[528]     574.83    4.25 235.78 314.55 534.52  875.07  3072 1.00
## y_rep[529]     273.35    2.08 113.62 150.16 251.96  421.61  2992 1.00
## y_rep[530]     279.48    2.07 114.03 157.32 257.83  427.74  3037 1.00
## y_rep[531]     186.08    1.50  79.78 102.97 171.10  287.39  2815 1.00
## y_rep[532]     410.51    3.14 171.12 226.37 377.66  634.24  2962 1.00
## y_rep[533]     416.12    3.14 170.01 228.40 386.16  646.84  2939 1.00
## y_rep[534]     206.80    1.57  86.30 115.47 190.17  317.99  3021 1.00
## y_rep[535]     283.17    2.31 117.45 151.76 264.39  435.74  2587 1.00
## y_rep[536]     661.92    5.21 279.79 368.25 612.87 1025.43  2880 1.00
## y_rep[537]     178.62    1.34  74.32  97.70 164.78  275.26  3088 1.00
## y_rep[538]     309.57    2.31 128.15 174.25 283.72  478.58  3081 1.00
## y_rep[539]     187.63    1.47  78.01 104.81 172.49  286.64  2819 1.00
## y_rep[540]     462.78    3.60 191.62 258.32 431.78  703.26  2840 1.00
## y_rep[541]     415.03    3.04 170.01 231.35 382.26  631.48  3137 1.00
## y_rep[542]     584.61    4.44 241.62 321.62 542.06  908.78  2955 1.00
## y_rep[543]     426.57    3.16 175.31 237.46 395.63  658.85  3075 1.00
## y_rep[544]     283.27    2.13 118.17 157.08 262.00  434.60  3084 1.00
## y_rep[545]     191.36    1.54  80.60 105.59 176.65  299.92  2733 1.00
## y_rep[546]     181.96    1.46  78.07 100.01 166.90  286.96  2865 1.00
## y_rep[547]     287.36    2.21 122.09 157.79 262.94  451.42  3047 1.00
## y_rep[548]     419.85    3.26 179.81 227.46 387.02  653.29  3047 1.00
## y_rep[549]     275.68    2.10 114.70 153.24 255.01  423.63  2992 1.00
## y_rep[550]     602.44    4.62 253.25 331.88 553.66  948.43  3010 1.00
## y_rep[551]     590.26    4.40 245.51 322.18 550.62  904.40  3106 1.00
## y_rep[552]     187.46    1.47  78.62 103.66 172.46  290.22  2844 1.00
## y_rep[553]     658.18    5.11 273.78 362.25 609.44 1014.13  2868 1.00
## y_rep[554]     313.14    2.32 131.03 173.05 287.88  489.54  3199 1.00
## y_rep[555]     202.11    1.69  83.89 111.33 186.29  311.67  2471 1.00
## y_rep[556]     281.75    2.10 113.92 156.70 260.87  431.84  2942 1.00
## y_rep[557]     189.93    1.47  78.77 105.38 174.36  292.53  2883 1.00
## y_rep[558]     408.99    3.20 171.50 221.85 379.97  636.04  2881 1.00
## y_rep[559]     205.88    1.54  85.75 114.02 190.53  319.38  3093 1.00
## y_rep[560]     182.44    1.47  78.56  98.45 167.97  284.03  2852 1.00
## y_rep[561]     584.37    4.66 241.15 324.49 542.59  889.91  2680 1.00
## y_rep[562]     596.73    4.77 253.47 327.37 549.45  915.88  2829 1.00
## y_rep[563]     313.66    2.38 130.50 172.35 289.78  488.28  3002 1.00
## y_rep[564]     425.09    3.22 173.36 241.29 390.94  652.16  2890 1.00
## y_rep[565]     274.35    2.15 117.82 148.03 252.95  422.60  2995 1.00
## y_rep[566]     184.79    1.56  79.17 100.30 170.56  283.74  2582 1.00
## y_rep[567]     281.18    2.24 118.73 155.16 259.44  438.17  2811 1.00
## y_rep[568]     185.18    1.46  78.46 102.06 169.15  285.93  2883 1.00
## y_rep[569]     574.82    4.64 241.89 319.25 523.14  893.26  2713 1.00
## y_rep[570]     284.82    2.10 115.99 160.74 260.74  435.50  3046 1.00
## y_rep[571]     422.67    3.13 175.02 235.50 389.27  649.58  3130 1.00
## y_rep[572]     591.82    4.49 242.87 329.07 550.89  898.95  2931 1.00
## y_rep[573]     284.66    2.25 119.22 158.21 263.68  432.30  2803 1.00
## y_rep[574]     426.17    3.42 183.61 230.45 390.02  657.33  2875 1.00
## y_rep[575]     464.74    3.56 196.53 255.72 426.89  718.33  3044 1.00
## y_rep[576]     204.70    1.54  85.25 111.07 188.89  317.40  3058 1.00
## y_rep[577]     273.44    2.10 113.03 151.64 255.31  421.65  2888 1.00
## y_rep[578]     311.24    2.41 129.33 169.94 288.26  482.52  2884 1.00
## y_rep[579]     652.26    5.07 277.71 353.36 600.82 1005.26  3002 1.00
## y_rep[580]     182.02    1.44  76.70 101.19 167.50  276.27  2828 1.00
## y_rep[581]     598.00    4.45 246.56 335.71 550.96  914.60  3065 1.00
## y_rep[582]     409.04    3.12 171.04 228.65 376.67  635.02  3011 1.00
## y_rep[583]     187.63    1.44  76.46 103.28 175.48  288.23  2801 1.00
## y_rep[584]     187.96    1.54  79.22 104.27 172.71  291.29  2654 1.00
## y_rep[585]     275.80    2.08 114.50 154.05 254.62  423.00  3042 1.00
## y_rep[586]     205.74    1.53  84.95 114.42 190.41  318.41  3067 1.00
## y_rep[587]     463.01    3.77 188.36 255.73 431.79  715.23  2499 1.00
## y_rep[588]     309.99    2.41 132.73 173.07 281.30  482.24  3029 1.00
## y_rep[589]     419.44    3.38 176.45 232.17 382.35  660.51  2731 1.00
## y_rep[590]     272.69    2.20 118.40 147.71 249.18  426.90  2907 1.00
## y_rep[591]     602.65    4.79 251.61 330.85 554.72  935.79  2761 1.00
## y_rep[592]     659.81    5.11 276.28 359.43 607.64 1017.83  2918 1.00
## y_rep[593]     286.08    2.21 120.43 160.80 261.05  441.61  2966 1.00
## y_rep[594]     423.38    3.09 175.46 234.39 390.17  645.67  3232 1.00
## y_rep[595]     180.76    1.37  77.26 100.40 164.71  282.67  3177 1.00
## y_rep[596]     186.47    1.60  76.96 102.27 172.18  287.45  2323 1.00
## y_rep[597]     592.59    4.51 245.86 330.29 545.42  908.52  2976 1.00
## y_rep[598]     587.00    4.32 243.04 322.63 542.23  903.20  3163 1.00
## y_rep[599]     462.32    3.65 192.58 258.64 427.43  724.49  2785 1.00
## y_rep[600]     205.89    1.56  86.66 113.72 189.74  319.20  3071 1.00
## y_rep[601]     602.48    4.61 249.58 330.73 556.64  919.13  2926 1.00
## y_rep[602]     287.96    2.27 120.83 157.31 265.81  442.23  2825 1.00
## y_rep[603]     186.88    1.46  78.14 102.33 171.81  286.71  2864 1.00
## y_rep[604]     275.78    2.22 118.01 150.90 254.26  424.07  2819 1.00
## y_rep[605]     425.50    3.18 173.48 235.65 395.36  651.05  2976 1.00
## y_rep[606]     653.10    5.12 276.51 353.33 605.98 1010.86  2921 1.00
## y_rep[607]     182.28    1.36  75.16 100.04 169.62  277.28  3070 1.00
## y_rep[608]     406.94    2.98 166.92 225.28 377.02  619.64  3133 1.00
## y_rep[609]     185.17    1.37  74.88 101.82 171.63  287.43  2970 1.00
## y_rep[610]     308.58    2.51 126.09 172.47 286.70  476.20  2533 1.00
## y_rep[611]     660.83    5.04 275.68 364.12 606.45 1020.82  2986 1.00
## y_rep[612]     280.68    2.33 120.45 156.69 255.42  430.66  2672 1.00
## y_rep[613]     205.88    1.52  83.22 115.31 190.53  314.79  2999 1.00
## y_rep[614]     593.18    4.52 242.42 330.19 547.98  903.41  2873 1.00
## y_rep[615]     424.70    3.29 179.40 234.01 391.88  654.06  2968 1.00
## y_rep[616]     418.35    3.22 172.03 230.80 387.33  648.48  2847 1.00
## y_rep[617]     286.19    2.29 120.60 158.23 261.18  442.63  2785 1.00
## y_rep[618]     310.73    2.37 126.05 175.56 287.58  477.72  2824 1.00
## y_rep[619]     605.13    4.88 252.40 337.31 555.84  939.98  2673 1.00
## y_rep[620]     183.19    1.38  76.41 101.34 169.15  279.76  3076 1.00
## y_rep[621]     183.78    1.36  74.62 102.84 170.81  280.20  3023 1.00
## y_rep[622]     463.15    3.52 189.55 258.27 433.37  708.64  2901 1.00
## y_rep[623]     187.57    1.48  78.66 103.90 172.27  289.37  2820 1.00
## y_rep[624]     308.87    2.28 123.82 171.83 285.52  475.63  2957 1.00
## y_rep[625]     662.24    5.12 280.61 375.62 605.89 1016.12  3008 1.00
## y_rep[626]     190.52    1.54  81.05 105.46 174.34  293.75  2766 1.00
## y_rep[627]     280.92    2.10 115.57 157.34 260.26  431.72  3015 1.00
## y_rep[628]     606.31    4.63 249.68 334.39 560.85  926.92  2911 1.00
## y_rep[629]     271.40    1.97 108.74 151.40 250.74  419.03  3061 1.00
## y_rep[630]     179.45    1.33  72.36 100.63 167.15  274.74  2947 1.00
## y_rep[631]     423.14    3.46 176.89 235.41 391.52  652.57  2610 1.00
## y_rep[632]     464.71    3.52 193.23 257.32 428.51  715.70  3015 1.00
## y_rep[633]     283.82    2.11 115.44 154.35 263.34  437.97  2989 1.00
## y_rep[634]     202.85    1.55  85.08 112.97 189.56  309.29  3009 1.00
## y_rep[635]     186.12    1.53  76.83 101.57 171.79  290.86  2520 1.00
## y_rep[636]     196.60    1.48  79.91 108.67 183.86  297.91  2917 1.00
## y_rep[637]     198.99    1.50  81.82 110.70 183.36  306.16  2987 1.00
## y_rep[638]     632.38    5.03 268.48 348.27 573.53  987.63  2847 1.00
## y_rep[639]     449.56    3.40 189.82 248.17 413.15  699.41  3110 1.00
## y_rep[640]     301.22    2.38 124.33 169.56 277.37  461.56  2740 1.00
## y_rep[641]     296.50    2.30 123.55 167.67 273.94  448.70  2879 1.00
## y_rep[642]     194.97    1.45  80.71 107.85 178.71  303.95  3112 1.00
## y_rep[643]     302.63    2.31 125.38 167.54 278.70  466.16  2937 1.00
## y_rep[644]     293.77    2.21 121.47 163.85 271.32  456.76  3023 1.00
## y_rep[645]     443.11    3.52 189.40 244.95 408.80  675.99  2901 1.00
## y_rep[646]     626.64    4.88 262.51 350.89 573.63  964.73  2895 1.00
## y_rep[647]     445.99    3.46 188.57 246.36 410.29  686.83  2970 1.00
## y_rep[648]     644.72    5.29 264.71 353.53 598.23  984.50  2499 1.00
## y_rep[649]     303.57    2.35 128.79 162.74 280.66  469.25  2994 1.00
## y_rep[650]     640.53    4.73 267.32 356.79 593.21  986.06  3191 1.00
## y_rep[651]     716.46    5.73 309.34 385.82 656.63 1115.30  2915 1.00
## y_rep[652]     655.94    5.09 275.31 360.48 605.80 1010.26  2930 1.00
## y_rep[653]     441.89    3.53 188.35 244.53 404.70  687.99  2840 1.00
## y_rep[654]     459.71    3.60 189.43 256.87 427.13  700.53  2777 1.00
## y_rep[655]     295.03    2.23 121.16 164.46 270.95  458.25  2951 1.00
## y_rep[656]     636.87    4.84 264.16 349.95 586.45  977.34  2974 1.00
## y_rep[657]     338.91    2.76 145.78 188.31 309.58  524.53  2796 1.00
## y_rep[658]     306.29    2.49 126.95 171.62 283.64  472.79  2598 1.00
## y_rep[659]     493.96    3.85 205.57 270.58 455.96  763.04  2849 1.00
## y_rep[660]     451.32    3.47 190.09 249.05 413.54  704.88  2996 1.00
## y_rep[661]     440.50    3.23 180.97 241.81 407.67  682.29  3134 1.00
## y_rep[662]     504.21    3.78 203.20 282.29 470.00  770.75  2886 1.00
## y_rep[663]     332.56    2.55 137.86 181.66 310.16  512.20  2933 1.00
## y_rep[664]     295.25    2.31 126.02 159.27 274.84  457.22  2965 1.00
## y_rep[665]     711.95    5.12 289.08 400.71 660.90 1082.23  3193 1.00
## y_rep[666]     643.24    5.30 271.85 352.37 595.66  994.72  2632 1.00
## y_rep[667]     193.18    1.51  80.42 108.33 178.70  296.51  2821 1.00
## y_rep[668]     306.65    2.36 130.41 166.84 281.21  477.30  3046 1.00
## y_rep[669]     655.22    5.18 279.83 349.69 605.79 1011.96  2921 1.00
## y_rep[670]     332.43    2.50 137.45 185.39 305.60  514.45  3011 1.00
## y_rep[671]     197.93    1.58  79.80 108.80 185.40  301.85  2560 1.00
## y_rep[672]     504.89    3.86 208.73 280.32 462.63  782.06  2917 1.00
## y_rep[673]     220.12    1.83  90.16 123.48 205.40  334.06  2438 1.00
## y_rep[674]     716.23    5.72 298.72 400.84 653.56 1096.56  2723 1.00
## y_rep[675]     470.12    3.72 199.99 255.39 432.94  733.03  2894 1.00
## y_rep[676]     200.03    1.57  83.64 108.02 183.45  312.28  2843 1.00
## y_rep[677]     200.50    1.51  84.63 109.87 185.25  307.90  3140 1.00
## y_rep[678]     501.76    3.77 208.18 280.41 464.49  779.44  3057 1.00
## y_rep[679]     306.07    2.49 127.60 169.74 280.37  472.40  2628 1.00
## y_rep[680]     198.24    1.48  82.09 109.82 183.21  302.32  3093 1.00
## y_rep[681]     651.71    4.91 272.21 357.90 598.85 1017.50  3068 1.00
## y_rep[682]     453.95    3.52 192.42 250.35 418.31  700.92  2983 1.00
## y_rep[683]     217.80    1.80  91.99 118.88 201.84  332.87  2609 1.00
## y_rep[684]     192.93    1.42  78.66 107.99 177.78  298.09  3070 1.00
## y_rep[685]     331.72    2.52 135.67 180.23 311.64  508.84  2905 1.00
## y_rep[686]     706.13    5.46 297.67 384.40 649.21 1098.70  2972 1.00
## y_rep[687]     194.40    1.50  82.48 106.50 178.51  304.18  3015 1.00
## y_rep[688]     654.62    5.01 276.26 359.91 599.77 1030.33  3046 1.00
## y_rep[689]     311.74    2.47 132.54 171.59 286.34  482.09  2884 1.00
## y_rep[690]     219.63    1.64  88.91 122.02 203.43  335.99  2932 1.00
## y_rep[691]     202.06    1.57  85.61 112.21 185.92  308.08  2984 1.00
## y_rep[692]     295.37    2.24 122.83 161.75 273.24  455.02  3014 1.00
## y_rep[693]     197.68    1.50  84.78 109.11 180.75  305.60  3182 1.00
## y_rep[694]     455.89    3.57 188.60 259.31 418.24  698.04  2790 1.00
## y_rep[695]     308.82    2.43 130.31 164.73 284.30  475.83  2881 1.00
## y_rep[696]     655.85    4.88 277.67 362.42 607.13 1014.91  3235 1.00
## y_rep[697]     194.80    1.46  80.10 108.51 181.95  297.38  3018 1.00
## y_rep[698]     218.16    1.60  89.41 122.62 201.54  335.00  3118 1.00
## y_rep[699]     294.40    2.23 122.12 163.14 270.69  458.31  2999 1.00
## y_rep[700]     462.35    3.70 189.86 253.31 426.91  708.01  2638 1.00
## y_rep[701]     309.38    2.32 128.40 170.18 284.68  483.85  3060 1.00
## y_rep[702]     205.65    1.69  86.59 114.14 190.48  316.96  2628 1.00
## y_rep[703]     200.03    1.54  82.50 111.65 185.05  310.79  2871 1.00
## y_rep[704]     331.27    2.44 136.05 182.05 308.51  508.23  3101 1.00
## y_rep[705]     334.26    2.60 144.02 185.46 302.78  527.26  3061 1.00
## y_rep[706]     301.09    2.29 126.62 165.33 278.50  473.39  3064 1.00
## y_rep[707]     196.65    1.49  81.17 108.43 183.25  302.45  2986 1.00
## y_rep[708]     193.80    1.49  78.77 108.00 181.43  291.04  2813 1.00
## y_rep[709]     201.21    1.57  83.53 109.86 187.84  309.11  2847 1.00
## y_rep[710]     650.37    5.11 270.03 351.69 599.78 1001.58  2792 1.00
## y_rep[711]     452.99    3.74 187.61 247.30 420.10  689.53  2517 1.00
## y_rep[712]     714.84    5.61 301.17 395.36 662.52 1075.16  2882 1.00
## y_rep[713]     218.25    1.72  90.25 123.78 200.69  334.41  2739 1.00
## y_rep[714]     295.48    2.34 124.86 161.64 269.99  455.72  2844 1.00
## y_rep[715]     497.88    3.71 202.64 282.63 461.59  758.84  2979 1.00
## y_rep[716]     307.94    2.37 128.27 173.40 281.69  477.38  2919 1.00
## y_rep[717]     297.55    2.34 120.37 166.88 275.45  451.92  2641 1.00
## y_rep[718]     455.79    3.49 191.20 247.47 422.37  696.91  3003 1.00
## y_rep[719]     310.54    2.29 127.47 173.17 287.62  483.27  3100 1.00
## y_rep[720]     206.93    1.56  86.53 113.83 190.96  314.80  3062 1.00
## y_rep[721]     500.94    3.69 206.43 278.32 461.00  779.95  3131 1.00
## y_rep[722]     341.54    2.68 142.91 191.10 315.32  521.53  2841 1.00
## y_rep[723]     227.43    1.71  90.75 126.52 212.84  345.25  2829 1.00
## y_rep[724]     461.75    3.45 192.31 255.13 425.28  718.65  3100 1.00
## y_rep[725]     620.57    4.66 255.36 351.91 571.14  961.42  3007 1.00
## y_rep[726]     201.49    1.50  83.49 109.30 186.61  307.03  3098 1.00
## y_rep[727]     209.39    1.60  85.56 115.88 194.59  323.41  2850 1.00
## y_rep[728]     684.60    5.34 285.58 374.37 631.02 1064.35  2861 1.00
## y_rep[729]     624.73    4.91 258.72 347.21 581.56  964.51  2779 1.00
## y_rep[730]     313.73    2.86 135.00 173.45 284.58  493.31  2228 1.00
## y_rep[731]     228.73    1.79  94.97 127.10 211.27  355.58  2804 1.00
## y_rep[732]     311.03    2.48 130.91 175.92 285.82  481.99  2777 1.00
## y_rep[733]     450.34    3.27 184.89 247.95 418.24  699.85  3190 1.00
## y_rep[734]     449.77    3.53 188.94 245.79 414.25  696.00  2857 1.00
## y_rep[735]     301.83    2.29 123.98 165.96 279.71  463.25  2918 1.00
## y_rep[736]     200.55    1.58  81.53 110.44 185.87  309.03  2660 1.00
## y_rep[737]     208.35    1.55  86.08 116.47 192.02  320.57  3082 1.00
## y_rep[738]     624.46    4.94 258.90 348.48 574.50  972.59  2744 1.00
## y_rep[739]     614.79    4.69 247.95 345.78 569.76  934.83  2792 1.00
## y_rep[740]     208.59    1.68  87.49 114.38 193.00  318.69  2713 1.00
## y_rep[741]     201.38    1.44  81.60 112.16 188.75  311.96  3217 1.00
## y_rep[742]     204.70    1.51  83.28 115.29 190.99  315.82  3034 1.00
## y_rep[743]     448.77    3.43 186.23 243.94 415.20  690.86  2944 1.00
## y_rep[744]     501.75    3.79 208.88 276.66 465.68  766.01  3032 1.00
## y_rep[745]     341.06    2.72 143.64 188.87 316.73  523.41  2796 1.00
## y_rep[746]     302.30    2.31 126.01 165.61 280.52  459.55  2976 1.00
## y_rep[747]     632.45    4.82 259.59 354.18 578.50  976.77  2899 1.00
## y_rep[748]     229.60    1.73  96.92 125.24 210.55  356.96  3150 1.00
## y_rep[749]     309.48    2.32 127.91 172.22 286.74  479.42  3035 1.00
## y_rep[750]     624.99    4.87 263.84 339.71 575.94  967.24  2936 1.00
## y_rep[751]     313.82    2.56 134.24 171.93 288.34  486.73  2752 1.00
## y_rep[752]     456.98    3.55 186.06 255.69 421.43  695.16  2745 1.00
## y_rep[753]     205.65    1.52  84.99 114.25 189.00  317.26  3128 1.00
## y_rep[754]     689.85    5.22 280.93 383.73 642.98 1052.93  2892 1.00
## y_rep[755]     230.53    1.93  97.24 125.41 211.91  359.11  2539 1.00
## y_rep[756]     632.82    5.07 264.73 346.19 584.63  969.74  2724 1.00
## y_rep[757]     337.52    2.57 136.32 193.86 311.00  511.44  2809 1.00
## y_rep[758]     300.02    2.43 125.47 168.18 278.54  455.23  2676 1.00
## y_rep[759]     309.14    2.33 127.58 171.10 286.71  471.73  3007 1.00
## y_rep[760]     446.31    3.41 187.87 245.96 412.33  691.03  3032 1.00
## y_rep[761]     201.92    1.55  84.86 110.37 185.18  311.52  3002 1.00
## y_rep[762]     607.84    4.52 247.67 337.07 563.53  935.17  3008 1.00
## y_rep[763]     451.69    3.31 183.19 253.44 417.46  687.28  3067 1.00
## y_rep[764]     686.99    5.06 283.27 384.73 632.53 1057.27  3129 1.00
## y_rep[765]     625.38    4.91 265.17 337.90 575.23  970.45  2912 1.00
## y_rep[766]     315.35    2.43 131.81 172.67 292.03  484.76  2949 1.00
## y_rep[767]     501.11    3.72 201.92 279.06 466.07  766.15  2952 1.00
## y_rep[768]     209.04    1.59  85.47 116.03 194.67  321.90  2891 1.00
## y_rep[769]     208.87    1.56  85.25 117.31 193.11  323.64  3002 1.00
## y_rep[770]     457.18    3.50 191.22 253.14 421.29  706.32  2980 1.00
## y_rep[771]     225.52    1.61  91.62 125.80 209.12  344.22  3249 1.00
## y_rep[772]     460.36    3.66 194.33 256.20 423.13  710.88  2827 1.00
## y_rep[773]     206.70    1.58  86.29 113.91 192.19  313.99  2996 1.00
## y_rep[774]     302.89    2.27 124.92 171.05 279.62  460.83  3034 1.00
## y_rep[775]     633.18    4.85 264.36 354.49 579.87  965.71  2975 1.00
## y_rep[776]     445.08    3.32 182.97 250.99 411.26  691.64  3043 1.00
## y_rep[777]     335.81    2.57 135.98 191.41 310.50  508.74  2794 1.00
## y_rep[778]     201.77    1.51  82.51 113.74 188.92  304.83  3000 1.00
## y_rep[779]     209.14    1.54  85.68 116.56 195.34  323.70  3093 1.00
## y_rep[780]     299.35    2.31 125.99 166.64 276.91  454.43  2970 1.00
## y_rep[781]     628.60    5.04 260.61 351.38 577.16  967.27  2672 1.00
## y_rep[782]     307.54    2.32 125.15 170.63 286.43  471.72  2922 1.00
## y_rep[783]     202.59    1.70  86.43 111.79 186.64  310.18  2597 1.00
## y_rep[784]     228.88    1.85  98.45 125.94 209.70  358.04  2847 1.00
## y_rep[785]     308.64    2.36 128.53 173.12 282.94  476.46  2967 1.00
## y_rep[786]     208.50    1.56  85.77 117.08 192.03  320.06  3011 1.00
## y_rep[787]     628.86    5.03 267.89 345.70 576.93  978.92  2835 1.00
## y_rep[788]     687.28    5.82 293.42 381.70 631.70 1059.29  2540 1.00
## y_rep[789]     297.60    2.29 125.54 164.04 272.34  457.11  3015 1.00
## y_rep[790]     206.39    1.59  88.15 112.09 189.49  322.34  3082 1.00
## y_rep[791]     443.69    3.42 183.51 249.43 410.19  685.01  2885 1.00
## y_rep[792]     441.57    3.55 184.24 243.33 408.47  688.32  2690 1.00
## y_rep[793]     500.44    3.75 207.13 273.70 467.89  771.23  3056 1.00
## y_rep[794]     339.93    2.62 140.15 189.84 314.46  510.17  2863 1.00
## y_rep[795]     230.28    1.75  95.10 130.52 211.33  354.78  2951 1.00
## y_rep[796]     502.86    4.04 205.96 277.97 465.00  779.35  2602 1.00
## y_rep[797]     311.70    2.20 123.05 174.17 289.96  477.99  3117 1.00
## y_rep[798]     633.12    4.52 260.70 343.10 587.91  969.77  3327 1.00
## y_rep[799]     210.88    1.62  88.61 116.72 194.31  327.69  3009 1.00
## y_rep[800]     460.48    3.46 190.81 253.62 428.06  716.55  3043 1.00
## y_rep[801]     207.89    1.56  86.31 114.65 192.17  317.28  3054 1.00
## y_rep[802]     455.07    3.65 192.46 252.43 420.02  689.88  2786 1.00
## y_rep[803]     610.34    4.60 252.24 340.47 567.95  941.66  3001 1.00
## y_rep[804]     624.88    4.59 255.65 347.77 575.72  967.24  3100 1.00
## y_rep[805]     338.21    2.61 142.46 186.77 309.43  516.85  2989 1.00
## y_rep[806]     447.27    3.48 188.37 245.88 410.81  684.11  2929 1.00
## y_rep[807]     200.20    1.50  81.19 110.73 187.63  308.49  2934 1.00
## y_rep[808]     308.32    2.27 125.05 172.20 286.20  475.28  3037 1.00
## y_rep[809]     301.69    2.25 122.05 170.70 280.66  455.93  2934 1.00
## y_rep[810]     300.11    2.28 127.44 163.60 275.53  467.17  3119 1.00
## y_rep[811]     436.40    3.39 183.65 240.59 396.00  673.03  2941 1.00
## y_rep[812]     309.41    2.42 127.16 169.58 285.12  479.58  2760 1.00
## y_rep[813]     628.20    4.71 256.93 349.35 583.20  960.69  2980 1.00
## y_rep[814]     636.57    4.74 261.27 349.30 588.22  980.55  3038 1.00
## y_rep[815]     198.55    1.51  81.39 109.98 183.93  302.28  2901 1.00
## y_rep[816]     688.17    5.21 288.72 381.52 631.54 1065.14  3073 1.00
## y_rep[817]     228.38    1.78  92.85 129.02 210.94  353.01  2715 1.00
## y_rep[818]     447.39    3.38 182.06 249.77 416.09  687.02  2909 1.00
## y_rep[819]     491.03    3.96 205.61 273.27 453.19  752.77  2692 1.00
## y_rep[820]     605.80    4.54 251.30 334.43 557.78  941.53  3065 1.00
## y_rep[821]     210.07    1.53  86.08 116.21 194.90  321.56  3172 1.00
## y_rep[822]     458.04    3.26 183.46 254.14 425.35  701.85  3167 1.00
## y_rep[823]     203.72    1.56  84.27 111.88 188.48  309.40  2934 1.00
## y_rep[824]     309.39    2.28 125.99 172.37 289.35  473.58  3053 1.00
## y_rep[825]     340.17    2.72 145.91 184.30 314.02  528.62  2874 1.00
## y_rep[826]     694.06    5.31 289.51 388.89 639.81 1078.41  2975 1.00
## y_rep[827]     227.91    1.87  96.82 127.37 208.08  346.90  2668 1.00
## y_rep[828]     304.11    2.25 125.21 169.22 281.95  466.91  3089 1.00
## y_rep[829]     455.14    3.36 187.07 254.15 418.77  706.72  3090 1.00
## y_rep[830]     300.28    2.38 124.73 166.43 277.48  463.04  2758 1.00
## y_rep[831]     462.80    3.50 195.08 256.31 424.98  717.19  3102 1.00
## y_rep[832]     501.24    3.81 210.67 280.16 463.02  770.70  3065 1.00
## y_rep[833]     210.18    1.61  88.25 115.42 193.05  324.50  3006 1.00
## y_rep[834]     313.24    2.32 130.58 171.97 291.22  476.48  3161 1.00
## y_rep[835]     336.54    2.45 135.69 192.22 311.79  515.53  3063 1.00
## y_rep[836]     200.95    1.57  84.20 110.24 185.25  309.07  2876 1.00
## y_rep[837]     633.40    5.00 264.15 348.97 588.08  980.32  2794 1.00
## y_rep[838]     623.98    4.80 260.85 346.38 571.89  950.14  2948 1.00
## y_rep[839]     204.40    1.64  86.04 114.33 187.35  314.58  2765 1.00
## y_rep[840]     311.27    2.31 128.13 173.26 287.26  474.75  3082 1.00
## y_rep[841]     230.86    1.76  95.92 126.67 213.57  356.48  2963 1.00
## y_rep[842]     631.38    4.71 259.25 348.05 587.49  983.93  3024 1.00
## y_rep[843]     635.95    4.91 270.49 350.21 584.99  993.91  3036 1.00
## y_rep[844]     335.02    2.52 138.19 185.87 309.52  507.66  3012 1.00
## y_rep[845]     454.46    3.56 185.43 254.85 418.41  695.44  2707 1.00
## y_rep[846]     604.31    4.42 246.85 340.33 557.92  927.28  3115 1.00
## y_rep[847]     200.71    1.50  82.33 112.39 184.61  309.38  3027 1.00
## y_rep[848]     436.66    3.34 181.31 240.10 399.89  674.60  2950 1.00
## y_rep[849]     697.12    5.11 288.43 390.08 643.53 1073.35  3189 1.00
## y_rep[850]     306.18    2.56 128.18 168.45 281.24  476.09  2511 1.00
## y_rep[851]     445.27    3.29 177.28 249.03 416.14  678.39  2902 1.00
## y_rep[852]     208.21    1.60  86.00 115.02 193.36  324.40  2875 1.00
## y_rep[853]     496.10    4.01 208.29 278.37 458.42  764.58  2699 1.00
## y_rep[854]     301.53    2.40 124.25 168.36 276.79  462.29  2678 1.00
## y_rep[855]     203.65    1.50  83.60 113.98 186.96  313.64  3094 1.00
## y_rep[856]     301.63    2.29 126.88 166.22 279.09  467.25  3061 1.00
## y_rep[857]     211.81    1.66  90.38 114.57 197.92  324.96  2966 1.00
## y_rep[858]     500.68    3.75 205.08 277.20 461.57  770.14  2994 1.00
## y_rep[859]     441.53    3.43 184.24 242.75 409.63  682.64  2886 1.00
## y_rep[860]     208.13    1.59  87.97 113.89 190.23  320.96  3080 1.00
## y_rep[861]     203.16    1.67  84.17 111.94 187.85  309.97  2554 1.00
## y_rep[862]     345.02    2.59 141.86 193.35 320.94  532.60  3011 1.00
## y_rep[863]     599.37    4.62 253.16 333.08 549.86  936.79  3003 1.00
## y_rep[864]     689.58    5.33 290.99 379.92 631.95 1086.18  2983 1.00
## y_rep[865]     224.98    1.77  93.19 123.69 206.49  347.70  2771 1.00
## y_rep[866]     498.63    3.73 208.61 275.85 460.96  759.24  3125 1.00
## y_rep[867]     618.60    4.43 251.74 340.75 571.10  961.98  3230 1.00
## y_rep[868]     211.12    1.58  87.09 116.12 194.94  327.38  3029 1.00
## y_rep[869]     316.53    2.42 134.19 176.55 290.64  482.12  3080 1.00
## y_rep[870]     636.57    4.95 268.44 350.93 582.95  984.27  2942 1.00
## y_rep[871]     687.43    5.45 290.80 379.02 633.69 1067.19  2850 1.00
## y_rep[872]     466.78    3.81 196.38 259.00 428.79  718.43  2654 1.00
## y_rep[873]     297.74    2.27 125.09 164.98 272.64  462.63  3034 1.00
## y_rep[874]     232.28    1.72  96.27 127.15 212.92  357.30  3125 1.00
## y_rep[875]     448.21    3.35 185.52 246.73 414.95  687.56  3058 1.00
## y_rep[876]     338.07    2.52 135.10 190.74 315.55  514.51  2881 1.00
## y_rep[877]     209.70    1.58  88.09 116.18 191.20  323.37  3097 1.00
## y_rep[878]     306.77    2.31 125.46 168.83 285.93  466.33  2946 1.00
## y_rep[879]     202.42    1.56  84.00 111.26 187.41  308.67  2917 1.00
## lp__           389.00    0.27   5.09 382.48 388.92  395.47   365 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:18:33 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod3_1, pars = "y_rep") 
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25), 
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

Let’s call the correlation matrix \(\mathbf{R}\):

\[\mathbf{R} = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}\] If the prior was:

\[\mathbf{R} \sim \text{LKJcorr(2)}\]

There were more divergent transitions.

If the prior was:

\[\mathbf{R} \sim \text{LKJcorr(4)}\]

There were less divergent transitions, see the model below.

McElreath: “So whatever is the LKJcorr distribution? What LKJcorr(2) does is define a weakly informative prior on \(\rho\) that is skeptical of extreme correlations near −1 or 1. You can think of it as a regularizing prior for correlations. This distribution has a single parameter, \(\eta\), that controls how skeptical the prior is of large correlations in the matrix. When we use LKJcorr(1), the prior is flat over all valid correlation matrices. When the value is greater than 1, such as the 2 we used above, then extreme correlations are less likely. To visualize this family of priors, it will help to sample random matrices from it and plot the distribution of correlations”

Rho2 <- rlkjcorr( 1e4 , K=2 , eta=2 )
Rho1 <- rlkjcorr( 1e4 , K=2 , eta=1 )
Rho4 <- rlkjcorr( 1e4 , K=2 , eta=4 )
plot_grid(dens(Rho1[,1,2],xlim=c(-1,1),xlab="correlation" ,ylim=c(0,1.2),main="eta=1"),
         dens(Rho2[,1,2] , xlab="correlation" ,xlim=c(-1,1),ylim=c(0,1.2),main="eta=2"),
         dens(Rho4[,1,2],xlim=c(-1,1),xlab="correlation" ,ylim=c(0,1.2),main="eta=4"))

3.3.1.2 mod3_2

Model with \(\mathbf{R} \sim \text{LKJcorr(4)}\)

mod3_2 = stan_model("mod3_2.stan")  
fit.mod3_2 <- sampling(mod3_2, data = data.list_mod3 , iter = 3000, chains = 2, cores = 2, control=list(max_treedepth=14,adapt_delta=0.99))   
print(fit.mod3_2, pars = c("beta_age","beta_age2",
                           "alpha","alpha_block", "sigma_block",
                           "Rho_prov","sigma_prov","alpha_prov","beta_prov","v_prov","SRS_prov",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9)) 
## Inference for Stan model: mod3_2.
## 2 chains, each with iter=3000; warmup=1500; thin=1; 
## post-warmup draws per chain=1500, total post-warmup draws=3000.
## 
##                 mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age        0.46    0.00 0.03  0.43  0.46  0.49   535 1.00
## beta_age2      -0.09    0.00 0.02 -0.11 -0.09 -0.07  1732 1.00
## alpha           5.85    0.01 0.10  5.76  5.85  5.93   197 1.00
## alpha_block[1] -0.05    0.00 0.08 -0.12 -0.05  0.02   314 1.00
## alpha_block[2] -0.03    0.00 0.07 -0.10 -0.02  0.04   320 1.00
## alpha_block[3] -0.01    0.00 0.07 -0.08 -0.01  0.06   322 1.00
## alpha_block[4]  0.07    0.00 0.07  0.01  0.07  0.15   344 1.00
## sigma_block     0.11    0.00 0.09  0.04  0.08  0.20   492 1.00
## Rho_prov[1,1]   1.00     NaN 0.00  1.00  1.00  1.00   NaN  NaN
## Rho_prov[1,2]  -0.08    0.01 0.35 -0.54 -0.08  0.39  1879 1.00
## Rho_prov[2,1]  -0.08    0.01 0.35 -0.54 -0.08  0.39  1879 1.00
## Rho_prov[2,2]   1.00    0.00 0.00  1.00  1.00  1.00  1961 1.00
## sigma_prov[1]   0.08    0.01 0.09  0.03  0.06  0.13   153 1.02
## sigma_prov[2]   0.03    0.00 0.04  0.00  0.02  0.07   442 1.01
## alpha_prov[1]   0.06    0.01 0.08  0.00  0.05  0.11   140 1.02
## alpha_prov[2]   0.01    0.01 0.08 -0.04  0.00  0.06   135 1.02
## alpha_prov[3]  -0.06    0.01 0.08 -0.13 -0.06 -0.01   164 1.01
## alpha_prov[4]   0.02    0.01 0.08 -0.04  0.01  0.07   147 1.02
## alpha_prov[5]   0.02    0.01 0.08 -0.04  0.01  0.07   144 1.01
## beta_prov[1]   -0.01    0.00 0.03 -0.04 -0.01  0.01   555 1.00
## beta_prov[2]    0.00    0.00 0.03 -0.03  0.00  0.03   547 1.00
## beta_prov[3]    0.01    0.00 0.03 -0.02  0.01  0.05   580 1.00
## beta_prov[4]    0.02    0.00 0.04 -0.01  0.01  0.07   466 1.00
## beta_prov[5]   -0.01    0.00 0.03 -0.04  0.00  0.02   687 1.00
## v_prov[1,1]     0.06    0.01 0.08  0.00  0.05  0.11   140 1.02
## v_prov[1,2]    -0.01    0.00 0.03 -0.04 -0.01  0.01   555 1.00
## v_prov[2,1]     0.01    0.01 0.08 -0.04  0.00  0.06   135 1.02
## v_prov[2,2]     0.00    0.00 0.03 -0.03  0.00  0.03   547 1.00
## v_prov[3,1]    -0.06    0.01 0.08 -0.13 -0.06 -0.01   164 1.01
## v_prov[3,2]     0.01    0.00 0.03 -0.02  0.01  0.05   580 1.00
## v_prov[4,1]     0.02    0.01 0.08 -0.04  0.01  0.07   147 1.02
## v_prov[4,2]     0.02    0.00 0.04 -0.01  0.01  0.07   466 1.00
## v_prov[5,1]     0.02    0.01 0.08 -0.04  0.01  0.07   144 1.01
## v_prov[5,2]    -0.01    0.00 0.03 -0.04  0.00  0.02   687 1.00
## SRS_prov[1,1]   0.01    0.01 0.09  0.00  0.00  0.02   200 1.01
## SRS_prov[1,2]   0.00    0.00 0.00  0.00  0.00  0.00  2944 1.00
## SRS_prov[2,1]   0.00    0.00 0.00  0.00  0.00  0.00  2944 1.00
## SRS_prov[2,2]   0.00    0.00 0.01  0.00  0.00  0.01   850 1.01
## sigma_y         0.40    0.00 0.01  0.38  0.40  0.41  1900 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:20:13 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod3_2, pars = "y_rep") 
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25), 
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

3.3.2 Non-centered parameterization

\[\begin{equation} \begin{aligned} h_{i} & \sim \text{LogNormal}(\mu_{i},\sigma_{i})\\[4pt] \mu_{i} & = \alpha + z_{\alpha_{BLOCK[b]}}\sigma_{\alpha_{BLOCK}} + z_{\alpha_{PROV[p]}}\sigma_{\alpha_{PROV}} + \beta_{age}age_{i} + \beta_{age2}age^{2}_{i} + z_{\gamma_{PROV[p]}}\sigma_{\gamma_{PROV}}age_{i} \\[4pt] \begin{bmatrix} z_{\alpha_{PROV[p]}} \\ z_{\gamma_{PROV[p]}} \end{bmatrix} & \sim \text{MVNormal}\left(\begin{bmatrix} 0 \\ 0 \end{bmatrix},\mathbf{R} \right) \\[4pt] \beta_{age} & \sim \text{LogNormal}(0,1) \\[4pt] \beta_{age2} & \sim \mathcal{N}(0,1)\\[4pt] \alpha & \sim \text{LogNormal}(0,1)\\[4pt] z_{\alpha_{BLOCK[p]}} & \sim \mathcal{N}(0,1)\\[4pt] \sigma_{\alpha_{BLOCK}} & \sim \text{Exponential}(1)\\[4pt] \sigma & \sim \text{Exponential}(1)\\[4pt] \sigma_{\gamma_{PROV}} & \sim \text{Exponential}(1)\\[4pt] \sigma_{\alpha_{PROV}} & \sim \text{Exponential}(1)\\[4pt] \mathbf{R}& \sim \text{LKJcorr(4)} \end{aligned} \end{equation}\]

3.3.2.1 Following McElreath

Statistical rethinking (first version): m13_6NC1 P405

Stan code here: Code model using non-centered parameterization.

mod3_3 = stan_model("mod3_3.stan") 
fit.mod3_3 <- sampling(mod3_3, data = data.list_mod3 , iter = 3000, chains = 2, cores = 2, control=list(max_treedepth=14,adapt_delta=0.99)) 
print(fit.mod3_3, pars = c("beta_age","beta_age2",
                           "alpha","z_alpha_block", "sigma_block",
                           "Rho_prov","sigma_prov","z_alpha_prov","z_beta_prov","v_prov",
                           "sigma_y"), probs = c(0.10, 0.5, 0.9)) 
## Inference for Stan model: mod3_3.
## 2 chains, each with iter=3000; warmup=1500; thin=1; 
## post-warmup draws per chain=1500, total post-warmup draws=3000.
## 
##                   mean se_mean   sd   10%   50%   90% n_eff Rhat
## beta_age          0.47    0.00 0.03  0.44  0.47  0.50  1105 1.00
## beta_age2        -0.09    0.00 0.02 -0.11 -0.09 -0.07  2963 1.00
## alpha             5.85    0.00 0.08  5.76  5.85  5.93  1212 1.01
## z_alpha_block[1] -0.61    0.02 0.64 -1.44 -0.59  0.20  1506 1.00
## z_alpha_block[2] -0.28    0.01 0.59 -1.05 -0.27  0.46  1540 1.00
## z_alpha_block[3] -0.11    0.01 0.59 -0.88 -0.09  0.62  1535 1.00
## z_alpha_block[4]  1.02    0.02 0.72  0.13  0.99  1.95  1509 1.00
## sigma_block       0.11    0.00 0.09  0.04  0.08  0.20   961 1.00
## Rho_prov[1,1]     1.00     NaN 0.00  1.00  1.00  1.00   NaN  NaN
## Rho_prov[1,2]    -0.06    0.01 0.32 -0.49 -0.06  0.38  2168 1.00
## Rho_prov[2,1]    -0.06    0.01 0.32 -0.49 -0.06  0.38  2168 1.00
## Rho_prov[2,2]     1.00    0.00 0.00  1.00  1.00  1.00  1105 1.00
## sigma_prov[1]     0.08    0.00 0.06  0.03  0.06  0.13   939 1.00
## sigma_prov[2]     0.03    0.00 0.03  0.00  0.02  0.07   814 1.01
## z_alpha_prov[1]   0.80    0.02 0.65  0.01  0.77  1.62  1629 1.00
## z_alpha_prov[2]   0.07    0.02 0.63 -0.73  0.07  0.87  1595 1.00
## z_alpha_prov[3]  -1.03    0.02 0.73 -1.97 -1.01 -0.12  1737 1.00
## z_alpha_prov[4]   0.13    0.01 0.68 -0.74  0.13  0.99  2277 1.00
## z_alpha_prov[5]   0.20    0.02 0.65 -0.59  0.19  0.99  1766 1.00
## z_beta_prov[1]   -0.37    0.02 0.78 -1.31 -0.39  0.59  2124 1.00
## z_beta_prov[2]   -0.05    0.02 0.80 -1.03 -0.08  0.97  2455 1.00
## z_beta_prov[3]    0.26    0.02 0.84 -0.82  0.26  1.32  2423 1.00
## z_beta_prov[4]    0.50    0.02 0.85 -0.58  0.52  1.58  2313 1.00
## z_beta_prov[5]   -0.28    0.02 0.81 -1.29 -0.29  0.71  2047 1.00
## v_prov[1,1]       0.80    0.02 0.65  0.01  0.77  1.62  1629 1.00
## v_prov[1,2]      -0.37    0.02 0.78 -1.31 -0.39  0.59  2124 1.00
## v_prov[2,1]       0.07    0.02 0.63 -0.73  0.07  0.87  1595 1.00
## v_prov[2,2]      -0.05    0.02 0.80 -1.03 -0.08  0.97  2455 1.00
## v_prov[3,1]      -1.03    0.02 0.73 -1.97 -1.01 -0.12  1737 1.00
## v_prov[3,2]       0.26    0.02 0.84 -0.82  0.26  1.32  2423 1.00
## v_prov[4,1]       0.13    0.01 0.68 -0.74  0.13  0.99  2277 1.00
## v_prov[4,2]       0.50    0.02 0.85 -0.58  0.52  1.58  2313 1.00
## v_prov[5,1]       0.20    0.02 0.65 -0.59  0.19  0.99  1766 1.00
## v_prov[5,2]      -0.28    0.02 0.81 -1.29 -0.29  0.71  2047 1.00
## sigma_y           0.40    0.00 0.01  0.38  0.40  0.41  3477 1.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:22:03 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod3_3, pars = "y_rep")  
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25), 
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

posterior_cp <- as.array(fit.mod3_3) 
np_cp <- nuts_params(fit.mod3_3)  
mcmc_trace(posterior_cp, pars =c( "alpha","sigma_prov[1]"), np = np_cp) + 
  xlab("Post-warmup iteration")
## No divergences to plot.

mcmc_pairs(posterior_cp, np = np_cp, pars = c("alpha","beta_age","beta_age2","sigma_y","sigma_prov[1]","sigma_prov[2]"),  
           off_diag_args = list(size = 1, alpha = 1/3),np_style = pairs_style_np(div_size=3, div_shape = 19))

Comment: I tried \(\alpha \sim \mathcal{N}(0,1)\), chains didn’t mix. Very high R-hat values and lots of divergent transitions.

3.3.2.2 Following Sorensen

Sorensen et al. 2016. Listing 8.

data.list_mod3_4 <- list(N=length(data$height),          # Number of observations
                  y=data$height,                         # Response variables
                  age=data$age.sc,                       # Tree age
                  nprov=length(unique(data$prov)),       # Number of provenances
                  nblock=length(unique(data$block)),     # Number of blocks
                  prov=as.numeric(data$prov),            # Provenances
                  bloc=as.numeric(data$block))           # Blocks
mod3_4 = stan_model("mod3_4.stan")  
fit.mod3_4 <- sampling(mod3_4, data = data.list_mod3_4 , iter = 2000, chains = 2, cores = 2, control=list(max_treedepth=14,adapt_delta=0.999))  
## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: The largest R-hat is 1.48, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
print(fit.mod3_4, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: mod3_4.
## 2 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=2000.
## 
##                  mean se_mean     sd    10%    50%     90% n_eff Rhat
## beta_age         0.46    0.00   0.03   0.43   0.46    0.50  1310 1.00
## beta_age2       -0.09    0.00   0.02  -0.11  -0.09   -0.07  2033 1.00
## alpha            1.06    0.07   1.27  -0.39   1.01    2.55   312 1.01
## sigma_y          0.40    0.00   0.01   0.38   0.40    0.41  1280 1.00
## sigma_block      3.94    1.15   2.98   0.08   3.89    7.49     7 1.37
## alpha_block[1]   3.62    1.06   2.26  -0.05   4.35    5.98     5 1.81
## alpha_block[2]   3.65    1.06   2.26  -0.03   4.38    6.00     5 1.81
## alpha_block[3]   3.66    1.06   2.26  -0.02   4.40    6.02     5 1.81
## alpha_block[4]   3.76    1.06   2.26   0.07   4.49    6.12     5 1.81
## z_u[1,1]         0.79    0.02   0.57   0.10   0.76    1.49   867 1.00
## z_u[1,2]         0.23    0.17   0.63  -0.62   0.29    0.99    14 1.17
## z_u[1,3]        -0.62    0.43   0.99  -1.80  -0.76    0.82     5 1.57
## z_u[1,4]         0.25    0.16   0.71  -0.71   0.34    1.09    19 1.12
## z_u[1,5]         0.34    0.13   0.63  -0.48   0.40    1.10    25 1.10
## z_u[2,1]        -0.31    0.02   0.86  -1.40  -0.33    0.81  1358 1.00
## z_u[2,2]        -0.02    0.02   0.83  -1.04  -0.02    0.98  1725 1.00
## z_u[2,3]         0.18    0.02   0.85  -0.90   0.18    1.26  1470 1.00
## z_u[2,4]         0.49    0.02   0.90  -0.64   0.49    1.60  1306 1.00
## z_u[2,5]        -0.24    0.02   0.87  -1.32  -0.27    0.88  1512 1.00
## sigma_u[1]       1.64    1.61   3.34   0.03   0.08    6.76     4 1.87
## sigma_u[2]       0.03    0.00   0.04   0.00   0.02    0.07   745 1.00
## L_u[1,1]         1.00     NaN   0.00   1.00   1.00    1.00   NaN  NaN
## L_u[1,2]         0.00     NaN   0.00   0.00   0.00    0.00   NaN  NaN
## L_u[2,1]        -0.10    0.01   0.45  -0.69  -0.12    0.55  1439 1.00
## L_u[2,2]         0.88    0.00   0.14   0.68   0.93    1.00  1031 1.00
## u[1,1]           1.17    1.13   2.18   0.01   0.06    5.27     4 2.49
## u[1,2]           1.13    1.13   2.18  -0.04   0.02    5.17     4 2.48
## u[1,3]           1.04    1.12   2.16  -0.13  -0.05    5.10     4 2.48
## u[1,4]           1.13    1.13   2.18  -0.05   0.02    5.22     4 2.49
## u[1,5]           1.14    1.13   2.18  -0.03   0.03    5.20     4 2.49
## u[2,1]          -0.01    0.00   0.03  -0.05  -0.01    0.01  1310 1.00
## u[2,2]           0.00    0.00   0.03  -0.03   0.00    0.03  1470 1.00
## u[2,3]           0.01    0.00   0.03  -0.02   0.00    0.04  1490 1.00
## u[2,4]           0.02    0.00   0.04  -0.01   0.01    0.07  1096 1.00
## u[2,5]          -0.01    0.00   0.03  -0.05   0.00    0.02  1217 1.00
## y_rep[1]       318.25    3.04 134.14 174.06 292.30  498.06  1943 1.00
## y_rep[2]       218.86    2.14  93.20 118.61 199.17  337.72  1894 1.00
## y_rep[3]       455.50    4.65 193.59 247.34 417.20  705.32  1733 1.00
## y_rep[4]       311.37    2.91 129.55 173.35 288.58  476.31  1981 1.00
## y_rep[5]       214.67    2.01  88.13 120.61 201.69  325.57  1916 1.00
## y_rep[6]       239.08    2.06  95.87 134.31 222.42  367.77  2176 1.00
## y_rep[7]       209.07    2.00  88.86 113.90 190.29  326.28  1975 1.00
## y_rep[8]       309.91    2.82 126.20 175.09 286.47  477.34  2009 1.00
## y_rep[9]       215.69    1.99  90.94 117.42 198.58  342.88  2078 1.00
## y_rep[10]      362.50    3.34 152.40 200.72 332.64  552.16  2085 1.00
## y_rep[11]      454.15    4.29 183.27 251.91 424.80  699.62  1822 1.00
## y_rep[12]      711.57    6.79 291.45 396.26 654.83 1099.31  1844 1.00
## y_rep[13]      326.72    3.13 134.30 180.77 302.84  496.64  1841 1.00
## y_rep[14]      217.71    2.07  91.81 120.00 202.07  331.81  1962 1.00
## y_rep[15]      318.43    2.93 132.39 179.13 295.07  487.76  2040 1.00
## y_rep[16]      523.17    4.90 216.62 292.97 481.92  818.92  1953 1.00
## y_rep[17]      631.54    6.62 283.47 339.77 576.80  977.82  1833 1.00
## y_rep[18]      212.14    2.15  89.63 114.68 195.85  332.47  1740 1.00
## y_rep[19]      241.15    2.31 102.37 133.38 219.97  372.95  1958 1.00
## y_rep[20]      216.18    1.94  89.70 120.74 199.89  330.16  2137 1.00
## y_rep[21]      637.94    5.55 260.28 358.84 587.21  983.72  2203 1.00
## y_rep[22]      356.43    3.28 146.72 197.65 330.83  546.97  1996 1.00
## y_rep[23]      211.59    2.11  86.56 116.42 196.99  322.23  1687 1.00
## y_rep[24]      237.15    2.16  95.77 133.05 222.01  359.25  1964 1.00
## y_rep[25]      330.40    3.11 139.95 182.39 302.63  510.17  2025 1.00
## y_rep[26]      457.22    4.57 191.68 250.95 421.03  710.36  1762 1.00
## y_rep[27]      314.98    3.12 132.87 174.41 289.61  488.96  1810 1.00
## y_rep[28]      471.97    4.28 194.10 263.56 433.49  721.38  2052 1.00
## y_rep[29]      519.05    4.95 211.81 290.69 479.52  798.48  1831 1.00
## y_rep[30]      712.82    6.84 295.75 398.38 653.94 1100.98  1871 1.00
## y_rep[31]      221.30    2.19  92.42 122.27 205.69  340.65  1780 1.00
## y_rep[32]      618.55    5.74 259.15 335.59 575.23  957.40  2041 1.00
## y_rep[33]      214.63    2.06  91.31 117.79 196.40  337.14  1973 1.00
## y_rep[34]      214.67    2.04  89.71 119.72 198.78  328.50  1935 1.00
## y_rep[35]      653.90    6.36 274.94 359.42 604.88 1017.07  1868 1.00
## y_rep[36]      616.71    5.52 253.63 340.31 571.89  952.24  2110 1.00
## y_rep[37]      516.78    5.18 225.58 286.29 465.68  805.23  1898 1.00
## y_rep[38]      308.25    2.93 128.82 172.05 283.24  478.54  1937 1.00
## y_rep[39]      364.50    3.53 153.75 200.16 336.65  565.24  1902 1.00
## y_rep[40]      649.51    6.20 269.22 354.05 604.75  998.51  1888 1.00
## y_rep[41]      466.68    4.19 191.94 259.33 432.81  709.37  2103 1.00
## y_rep[42]      321.80    3.29 134.82 176.83 296.82  496.03  1684 1.00
## y_rep[43]      444.64    4.28 190.06 249.10 405.64  679.43  1976 1.00
## y_rep[44]      209.02    1.91  84.22 114.49 195.14  321.77  1942 1.00
## y_rep[45]      242.34    2.31 104.17 133.66 221.79  376.46  2036 1.00
## y_rep[46]      712.74    6.73 301.36 387.18 666.35 1095.20  2003 1.00
## y_rep[47]      319.54    3.10 129.74 175.87 295.92  489.91  1748 1.00
## y_rep[48]      474.22    4.55 197.90 261.07 441.62  724.91  1888 1.00
## y_rep[49]      324.66    3.01 133.33 178.06 297.83  505.28  1964 1.00
## y_rep[50]      521.45    4.83 220.20 294.83 474.66  815.58  2077 1.00
## y_rep[51]      471.15    4.27 189.43 266.35 441.12  718.32  1965 1.00
## y_rep[52]      617.95    6.51 258.01 346.75 563.05  974.27  1573 1.00
## y_rep[53]      237.95    2.48 100.79 133.82 221.39  364.29  1646 1.00
## y_rep[54]      719.31    6.68 293.74 407.06 661.24 1101.74  1936 1.00
## y_rep[55]      217.48    2.03  90.27 120.43 199.28  338.47  1986 1.00
## y_rep[56]      309.99    2.97 129.91 171.28 285.61  468.27  1909 1.00
## y_rep[57]      215.73    1.92  89.39 119.70 198.82  331.16  2178 1.00
## y_rep[58]      207.62    1.92  85.85 115.94 190.91  320.08  1999 1.00
## y_rep[59]      361.55    3.30 148.74 200.60 334.72  555.76  2035 1.00
## y_rep[60]      648.94    6.27 273.02 360.86 590.87  990.67  1894 1.00
## y_rep[61]      449.22    4.20 189.38 249.65 414.75  696.37  2031 1.00
## y_rep[62]      316.29    2.99 129.15 173.79 294.80  482.27  1868 1.00
## y_rep[63]      319.23    2.90 127.70 178.48 295.61  483.91  1936 1.00
## y_rep[64]      311.50    2.82 127.23 171.48 287.13  479.92  2040 1.00
## y_rep[65]      210.49    2.05  88.79 116.00 192.97  324.55  1871 1.00
## y_rep[66]      647.09    6.21 271.08 355.22 594.25  995.09  1908 1.00
## y_rep[67]      214.21    1.98  88.03 117.61 198.71  321.49  1969 1.00
## y_rep[68]      516.08    4.91 213.50 287.24 483.99  774.03  1893 1.00
## y_rep[69]      709.15    6.52 292.30 392.77 653.89 1102.38  2009 1.00
## y_rep[70]      472.29    4.26 195.91 264.39 435.73  720.39  2113 1.00
## y_rep[71]      447.51    4.01 176.50 253.24 417.30  674.15  1937 1.00
## y_rep[72]      220.39    2.12  91.86 122.15 204.42  334.73  1876 1.00
## y_rep[73]      240.09    2.34 100.05 134.26 218.77  374.36  1832 1.00
## y_rep[74]      359.64    3.45 148.38 198.85 333.74  557.23  1848 1.00
## y_rep[75]      629.18    6.05 266.92 349.28 582.57  951.11  1947 1.00
## y_rep[76]      326.82    3.05 135.72 179.91 299.81  507.91  1974 1.00
## y_rep[77]      448.47    4.17 187.75 246.62 416.19  687.87  2023 1.00
## y_rep[78]      207.71    1.95  84.50 116.33 193.04  317.03  1872 1.00
## y_rep[79]      241.70    2.27  99.70 134.42 222.74  371.65  1922 1.00
## y_rep[80]      310.23    2.77 123.99 176.90 289.91  466.34  2000 1.00
## y_rep[81]      214.74    1.94  86.15 119.72 200.75  323.03  1972 1.00
## y_rep[82]      213.79    2.03  87.36 119.73 197.46  329.42  1857 1.00
## y_rep[83]      530.83    4.91 226.34 290.86 492.00  810.67  2121 1.00
## y_rep[84]      632.15    5.96 272.83 343.43 579.93  987.80  2093 1.00
## y_rep[85]      316.68    2.85 127.07 177.44 295.01  478.60  1994 1.00
## y_rep[86]      476.08    4.36 197.45 263.32 441.55  733.78  2054 1.00
## y_rep[87]      728.54    7.10 313.30 396.89 674.07 1126.64  1946 1.00
## y_rep[88]      359.77    3.41 149.81 199.24 332.23  557.06  1935 1.00
## y_rep[89]      449.68    4.26 189.74 250.67 410.40  697.31  1983 1.00
## y_rep[90]      617.65    5.63 251.25 346.99 571.47  942.13  1993 1.00
## y_rep[91]      244.79    2.33 101.17 135.46 226.26  373.29  1878 1.00
## y_rep[92]      525.73    4.88 217.10 293.04 489.02  807.45  1978 1.00
## y_rep[93]      208.57    1.85  83.43 115.43 193.66  322.93  2024 1.00
## y_rep[94]      219.34    2.11  92.44 120.59 202.12  333.95  1928 1.00
## y_rep[95]      310.79    3.01 131.30 172.17 287.83  477.76  1906 1.00
## y_rep[96]      322.70    3.09 134.48 179.21 297.57  498.75  1898 1.00
## y_rep[97]      720.91    6.92 308.26 391.17 664.63 1115.32  1982 1.00
## y_rep[98]      361.32    3.59 153.96 198.43 331.21  563.28  1838 1.00
## y_rep[99]      215.02    1.89  85.54 123.39 200.53  323.81  2048 1.00
## y_rep[100]     358.83    3.26 146.92 198.98 331.24  551.86  2029 1.00
## y_rep[101]     620.10    5.50 255.61 341.60 572.75  949.87  2158 1.00
## y_rep[102]     472.39    4.46 199.84 264.83 432.95  729.72  2006 1.00
## y_rep[103]     635.02    6.30 263.90 350.27 587.13  979.05  1753 1.00
## y_rep[104]     322.51    2.94 133.43 180.03 302.84  485.64  2059 1.00
## y_rep[105]     207.14    2.00  85.09 115.45 192.81  315.05  1803 1.00
## y_rep[106]     312.33    2.90 129.23 173.56 288.68  489.67  1992 1.00
## y_rep[107]     717.51    6.62 302.23 393.69 658.70 1132.47  2085 1.00
## y_rep[108]     237.23    2.16  98.95 130.37 217.76  370.62  2098 1.00
## y_rep[109]     320.99    2.81 131.06 176.28 297.87  486.59  2175 1.00
## y_rep[110]     454.93    4.28 191.21 250.99 418.06  700.66  1993 1.00
## y_rep[111]     214.87    1.97  88.34 119.23 196.57  329.23  2010 1.00
## y_rep[112]     653.61    6.33 277.84 354.69 601.79 1010.96  1926 1.00
## y_rep[113]     518.77    4.74 210.65 294.37 485.08  780.49  1979 1.01
## y_rep[114]     219.74    2.17  92.34 122.16 204.14  337.15  1808 1.00
## y_rep[115]     462.68    4.70 190.50 258.93 428.45  709.98  1641 1.00
## y_rep[116]     358.43    3.40 150.04 200.14 335.49  551.55  1949 1.00
## y_rep[117]     243.44    2.39 104.64 133.57 222.37  385.13  1923 1.00
## y_rep[118]     321.16    3.26 137.21 177.39 296.34  491.96  1774 1.00
## y_rep[119]     644.41    5.87 257.71 362.14 598.30  987.52  1930 1.00
## y_rep[120]     465.86    4.48 196.56 250.84 432.42  720.90  1928 1.00
## y_rep[121]     520.99    5.06 218.86 290.87 480.41  811.67  1870 1.00
## y_rep[122]     211.56    2.02  89.29 117.76 194.88  330.44  1952 1.00
## y_rep[123]     713.70    6.48 300.65 400.20 648.72 1102.75  2155 1.00
## y_rep[124]     307.58    2.82 128.10 172.53 283.42  470.82  2062 1.00
## y_rep[125]     216.37    2.25  89.45 121.36 200.68  330.42  1583 1.00
## y_rep[126]     218.06    2.13  95.46 118.19 199.19  343.67  2015 1.00
## y_rep[127]     239.23    2.07  93.48 137.51 225.09  362.74  2047 1.00
## y_rep[128]     715.92    6.94 306.49 394.97 662.89 1095.95  1953 1.00
## y_rep[129]     458.43    4.20 189.61 251.32 424.64  704.81  2039 1.00
## y_rep[130]     210.35    2.05  89.06 119.12 191.37  323.05  1886 1.00
## y_rep[131]     321.71    2.93 133.73 177.23 295.67  504.35  2082 1.00
## y_rep[132]     356.11    3.26 152.42 194.29 323.77  549.90  2183 1.00
## y_rep[133]     307.83    2.99 126.93 171.41 282.30  471.23  1797 1.00
## y_rep[134]     323.75    3.06 132.02 179.09 300.82  493.99  1863 1.00
## y_rep[135]     214.05    2.12  90.70 117.91 197.04  326.86  1832 1.00
## y_rep[136]     526.42    5.04 213.68 291.52 489.10  801.27  1797 1.00
## y_rep[137]     646.35    6.03 255.75 363.08 600.31  981.78  1800 1.00
## y_rep[138]     468.02    4.08 190.52 263.67 428.48  726.18  2176 1.00
## y_rep[139]     220.19    2.08  92.24 119.23 202.87  340.23  1964 1.00
## y_rep[140]     639.36    5.93 269.51 353.30 588.10  984.49  2069 1.00
## y_rep[141]     353.11    3.55 147.00 194.54 330.71  532.51  1711 1.00
## y_rep[142]     322.66    3.03 128.70 184.55 299.46  492.17  1810 1.00
## y_rep[143]     639.46    5.90 260.52 365.00 590.07  971.36  1950 1.00
## y_rep[144]     216.57    2.08  92.36 120.32 197.85  335.45  1978 1.00
## y_rep[145]     208.98    1.94  88.63 115.75 191.07  320.79  2095 1.00
## y_rep[146]     320.36    2.78 131.79 178.89 293.06  488.53  2253 1.00
## y_rep[147]     442.54    3.99 179.67 245.17 413.10  676.22  2024 1.00
## y_rep[148]     237.40    2.13  93.87 132.76 221.69  356.01  1934 1.00
## y_rep[149]     724.98    6.51 299.98 399.46 669.43 1114.90  2121 1.00
## y_rep[150]     308.97    2.96 125.80 174.43 286.84  476.14  1805 1.00
## y_rep[151]     467.63    4.35 192.40 253.65 431.49  711.69  1952 1.00
## y_rep[152]     216.53    2.11  90.96 118.89 199.39  334.61  1860 1.00
## y_rep[153]     613.60    5.64 245.00 341.33 571.54  945.48  1886 1.00
## y_rep[154]     520.53    5.01 218.62 281.65 479.65  820.23  1906 1.00
## y_rep[155]     362.97    3.54 151.78 199.07 333.80  565.05  1834 1.00
## y_rep[156]     215.42    2.01  88.79 118.95 199.21  333.39  1956 1.00
## y_rep[157]     659.82    6.14 278.78 354.77 612.60  999.53  2061 1.00
## y_rep[158]     213.76    2.09  90.10 120.10 196.72  331.23  1850 1.00
## y_rep[159]     722.70    7.05 297.43 405.94 672.20 1112.92  1779 1.00
## y_rep[160]     310.10    2.84 128.76 170.56 285.47  483.11  2055 1.00
## y_rep[161]     458.09    4.25 190.55 255.23 423.27  703.73  2008 1.00
## y_rep[162]     475.98    4.65 204.63 259.72 436.37  738.89  1936 1.00
## y_rep[163]     241.41    2.24  99.37 131.92 223.85  368.29  1975 1.00
## y_rep[164]     620.05    5.53 257.35 345.14 570.38  964.76  2169 1.00
## y_rep[165]     527.89    5.27 221.90 291.06 486.59  812.98  1772 1.00
## y_rep[166]     322.56    2.83 129.15 182.54 300.28  490.96  2077 1.00
## y_rep[167]     211.34    2.12  87.55 118.02 194.95  329.03  1700 1.00
## y_rep[168]     312.77    2.86 130.68 170.19 289.07  478.97  2088 1.00
## y_rep[169]     209.08    1.99  84.54 114.27 194.58  323.63  1798 1.00
## y_rep[170]     219.68    1.96  89.02 118.72 204.96  333.57  2060 1.00
## y_rep[171]     641.09    5.87 262.82 360.10 594.73  966.73  2002 1.00
## y_rep[172]     635.18    5.77 254.37 354.24 590.83  973.13  1944 1.00
## y_rep[173]     214.88    1.92  86.71 121.18 200.44  326.04  2043 1.00
## y_rep[174]     240.05    2.24  96.78 133.76 222.93  366.74  1874 1.00
## y_rep[175]     468.40    4.24 189.51 257.56 431.68  715.76  1995 1.00
## y_rep[176]     319.38    3.09 130.54 179.51 294.45  486.78  1789 1.00
## y_rep[177]     305.82    2.72 122.12 172.12 286.03  465.16  2017 1.00
## y_rep[178]     319.84    3.13 138.36 176.67 292.98  495.06  1949 1.00
## y_rep[179]     469.43    4.34 193.61 266.77 429.89  721.73  1987 1.00
## y_rep[180]     361.21    3.54 156.15 198.74 329.89  555.92  1946 1.00
## y_rep[181]     215.58    2.07  90.14 116.10 199.80  330.62  1900 1.00
## y_rep[182]     657.78    6.65 275.90 371.31 601.48 1005.69  1723 1.00
## y_rep[183]     477.21    4.38 194.19 265.46 441.48  732.20  1967 1.00
## y_rep[184]     218.74    2.10  92.37 119.29 198.81  341.07  1930 1.00
## y_rep[185]     237.58    2.12  95.77 130.74 220.63  360.76  2046 1.00
## y_rep[186]     324.27    3.09 134.28 180.31 299.42  502.26  1884 1.00
## y_rep[187]     208.81    1.92  84.02 117.73 194.18  313.29  1919 1.00
## y_rep[188]     358.20    3.27 147.59 197.51 330.93  549.53  2033 1.00
## y_rep[189]     239.97    2.15 101.07 130.22 221.96  372.82  2216 1.00
## y_rep[190]     207.85    1.93  87.70 113.76 193.34  317.79  2057 1.00
## y_rep[191]     657.61    6.22 275.20 364.44 608.92 1025.74  1957 1.00
## y_rep[192]     319.80    2.76 127.65 181.16 299.74  487.02  2143 1.00
## y_rep[193]     311.34    2.81 124.23 177.75 285.48  472.72  1958 1.00
## y_rep[194]     635.95    6.27 266.91 350.74 597.96  976.23  1812 1.00
## y_rep[195]     218.44    2.25  95.39 119.70 199.64  337.34  1792 1.00
## y_rep[196]     320.44    3.01 132.78 175.62 294.07  497.67  1940 1.00
## y_rep[197]     215.77    2.09  93.71 118.53 198.37  328.18  2020 1.00
## y_rep[198]     468.21    4.28 192.19 258.65 437.01  715.73  2017 1.00
## y_rep[199]     723.95    7.02 306.12 392.97 665.33 1130.46  1902 1.00
## y_rep[200]     515.63    4.91 213.65 282.05 473.65  799.72  1896 1.00
## y_rep[201]     470.40    4.55 196.02 260.91 433.39  729.99  1858 1.00
## y_rep[202]     352.93    3.39 145.32 197.42 324.65  540.73  1842 1.00
## y_rep[203]     324.89    3.12 136.59 180.18 298.66  500.49  1919 1.00
## y_rep[204]     451.95    4.67 192.49 253.62 418.24  682.54  1701 1.00
## y_rep[205]     236.54    2.45  97.98 130.08 220.47  356.70  1596 1.00
## y_rep[206]     463.01    4.29 189.13 254.93 429.54  722.57  1945 1.00
## y_rep[207]     355.09    3.34 147.01 193.90 329.43  550.12  1935 1.00
## y_rep[208]     216.45    2.04  88.39 122.20 200.10  330.08  1872 1.00
## y_rep[209]     311.54    2.94 129.29 172.18 284.85  487.63  1930 1.00
## y_rep[210]     645.33    6.03 260.37 354.55 605.79  995.49  1863 1.00
## y_rep[211]     637.64    6.04 269.47 350.40 591.70  969.49  1987 1.00
## y_rep[212]     514.05    4.64 206.89 282.53 484.13  778.24  1988 1.00
## y_rep[213]     220.43    2.06  89.00 123.21 204.45  336.74  1869 1.00
## y_rep[214]     476.42    4.52 199.89 264.71 442.41  727.93  1956 1.00
## y_rep[215]     207.07    1.93  85.14 116.40 189.38  322.52  1955 1.00
## y_rep[216]     323.61    3.52 135.86 180.05 297.14  502.60  1491 1.00
## y_rep[217]     627.64    5.65 256.62 352.35 584.24  952.04  2064 1.00
## y_rep[218]     619.46    5.98 261.45 343.13 568.43  944.18  1910 1.00
## y_rep[219]     206.86    1.90  85.62 113.38 191.57  321.57  2030 1.00
## y_rep[220]     466.51    4.33 198.12 247.87 426.10  716.28  2090 1.00
## y_rep[221]     323.68    3.10 136.58 186.38 296.30  504.64  1947 1.00
## y_rep[222]     220.11    2.10  92.22 123.26 203.41  331.87  1937 1.00
## y_rep[223]     308.43    2.89 127.60 171.15 284.70  471.53  1947 1.00
## y_rep[224]     357.41    3.36 143.82 204.03 332.38  542.85  1834 1.00
## y_rep[225]     459.55    4.34 194.10 257.15 420.02  710.36  2004 1.00
## y_rep[226]     215.19    2.07  92.37 118.60 197.66  333.29  1992 1.00
## y_rep[227]     239.92    2.32 102.92 129.18 219.47  378.98  1962 1.00
## y_rep[228]     468.56    4.27 187.69 260.96 440.41  719.06  1929 1.00
## y_rep[229]     638.33    6.12 261.65 356.85 587.83  975.82  1828 1.00
## y_rep[230]     325.69    2.96 134.41 180.59 301.08  500.28  2063 1.00
## y_rep[231]     650.89    6.42 268.29 356.27 606.53  974.62  1748 1.00
## y_rep[232]     320.90    3.10 132.85 177.53 296.41  495.18  1836 1.00
## y_rep[233]     521.48    5.17 218.37 286.12 479.41  810.03  1782 1.00
## y_rep[234]     206.77    1.88  85.01 112.72 192.13  314.38  2054 1.00
## y_rep[235]     215.08    2.00  90.80 117.85 198.84  331.78  2057 1.00
## y_rep[236]     466.20    4.32 193.42 260.04 427.56  720.57  2004 1.00
## y_rep[237]     216.95    2.02  90.22 119.37 198.95  336.29  2003 1.00
## y_rep[238]     240.73    2.14  99.10 134.47 223.36  371.40  2142 1.00
## y_rep[239]     309.67    2.81 124.16 169.94 289.02  479.42  1955 1.00
## y_rep[240]     359.84    3.68 157.39 197.11 326.85  566.05  1834 1.00
## y_rep[241]     615.81    5.73 252.73 342.32 571.48  942.27  1942 1.00
## y_rep[242]     449.98    4.07 182.16 252.36 416.35  707.51  2002 1.00
## y_rep[243]     317.23    2.87 127.61 178.22 294.51  486.04  1973 1.00
## y_rep[244]     642.54    6.14 274.40 357.79 590.51  976.74  1999 1.00
## y_rep[245]     716.67    7.05 300.05 409.04 652.95 1128.74  1810 1.00
## y_rep[246]     516.46    4.85 205.07 290.71 480.83  788.63  1785 1.00
## y_rep[247]     323.45    3.08 134.05 172.99 299.85  501.80  1897 1.00
## y_rep[248]     728.10    6.97 307.86 395.64 676.19 1131.30  1950 1.00
## y_rep[249]     215.34    1.98  88.58 121.39 198.65  330.88  2010 1.00
## y_rep[250]     650.22    6.52 274.87 354.25 601.08 1002.10  1777 1.00
## y_rep[251]     208.81    2.00  87.09 116.02 192.01  320.30  1902 1.00
## y_rep[252]     329.08    3.11 138.09 176.16 305.75  509.75  1968 1.00
## y_rep[253]     472.54    4.55 202.98 259.19 436.77  722.75  1993 1.00
## y_rep[254]     469.16    4.64 199.82 256.27 432.63  727.44  1857 1.00
## y_rep[255]     303.88    2.83 126.20 167.42 281.45  466.33  1985 1.00
## y_rep[256]     221.17    2.09  92.17 122.57 203.48  340.17  1954 1.00
## y_rep[257]     633.22    6.33 266.98 341.98 582.07  971.24  1780 1.00
## y_rep[258]     622.38    6.13 268.81 341.02 567.22  967.04  1921 1.00
## y_rep[259]     241.87    2.34 103.01 134.28 221.73  369.82  1932 1.00
## y_rep[260]     455.71    4.17 185.21 254.75 424.69  690.35  1975 1.00
## y_rep[261]     356.93    3.23 146.16 202.98 330.31  549.88  2046 1.00
## y_rep[262]     469.19    4.92 195.86 263.44 432.70  724.38  1583 1.00
## y_rep[263]     642.78    6.14 274.46 354.14 588.30 1000.94  2000 1.00
## y_rep[264]     205.33    1.99  86.39 115.94 187.08  316.90  1888 1.00
## y_rep[265]     359.18    4.15 152.50 195.04 330.34  553.76  1351 1.00
## y_rep[266]     210.52    2.04  82.99 118.70 197.67  317.36  1647 1.00
## y_rep[267]     474.85    4.41 203.93 263.45 435.97  734.15  2134 1.00
## y_rep[268]     527.85    4.73 214.27 291.03 491.29  808.28  2055 1.00
## y_rep[269]     309.36    2.95 125.02 169.20 287.84  474.94  1790 1.00
## y_rep[270]     243.92    2.24 102.07 134.62 225.02  370.73  2070 1.00
## y_rep[271]     717.46    6.82 291.36 396.50 669.95 1089.67  1823 1.00
## y_rep[272]     217.36    2.13  91.69 119.29 200.87  332.85  1858 1.00
## y_rep[273]     649.39    6.30 266.12 360.41 599.37  992.05  1784 1.00
## y_rep[274]     314.87    2.87 129.23 177.24 292.86  479.45  2034 1.00
## y_rep[275]     325.17    3.16 134.52 181.12 303.46  504.34  1817 1.00
## y_rep[276]     209.88    2.07  84.99 119.01 194.81  322.22  1693 1.00
## y_rep[277]     358.07    3.53 148.78 199.30 333.16  548.23  1777 1.00
## y_rep[278]     238.51    2.18  97.98 134.91 219.15  366.01  2018 1.00
## y_rep[279]     620.76    5.87 260.91 345.26 571.41  950.71  1972 1.00
## y_rep[280]     217.06    2.01  92.10 118.51 200.87  338.54  2098 1.00
## y_rep[281]     311.08    3.23 129.84 174.59 283.46  485.30  1612 1.00
## y_rep[282]     327.27    3.05 134.96 180.23 301.90  510.03  1962 1.00
## y_rep[283]     322.51    3.08 137.23 176.82 294.17  511.09  1991 1.00
## y_rep[284]     451.68    4.18 189.38 246.41 415.71  695.73  2050 1.00
## y_rep[285]     214.72    2.09  90.77 119.33 197.09  337.21  1879 1.00
## y_rep[286]     718.75    6.99 305.25 398.78 656.69 1115.95  1905 1.00
## y_rep[287]     522.81    4.64 208.76 296.61 484.49  795.56  2024 1.00
## y_rep[288]     458.67    4.42 188.46 257.81 427.94  700.41  1819 1.00
## y_rep[289]     467.52    4.30 194.43 255.14 430.91  712.80  2041 1.00
## y_rep[290]     218.10    1.99  89.21 121.85 202.22  329.75  2001 1.00
## y_rep[291]     472.58    4.43 194.85 261.40 435.25  740.07  1938 1.00
## y_rep[292]     618.30    5.71 256.65 340.91 570.30  958.73  2022 1.00
## y_rep[293]     640.25    6.16 269.24 352.88 591.86  991.10  1909 1.00
## y_rep[294]     210.89    1.97  88.99 117.10 193.94  327.87  2036 1.00
## y_rep[295]     324.64    3.15 134.52 181.15 298.55  493.59  1824 1.00
## y_rep[296]     311.35    2.88 130.93 170.70 287.95  478.88  2073 1.00
## y_rep[297]     213.07    1.91  88.82 118.70 197.90  329.93  2159 1.00
## y_rep[298]     512.28    4.37 201.93 289.44 478.27  774.98  2138 1.00
## y_rep[299]     647.51    5.94 275.51 364.76 599.28  984.70  2149 1.00
## y_rep[300]     318.13    2.89 129.02 174.79 297.03  486.54  1986 1.00
## y_rep[301]     703.86    6.67 287.59 395.78 654.77 1071.71  1859 1.00
## y_rep[302]     355.01    3.54 148.74 199.15 322.05  546.69  1764 1.00
## y_rep[303]     242.57    2.36 101.32 132.81 225.13  373.58  1841 1.00
## y_rep[304]     199.82    1.93  84.78 111.66 184.53  303.43  1923 1.00
## y_rep[305]     304.59    2.93 126.38 169.28 281.14  462.44  1866 1.00
## y_rep[306]     203.02    1.83  81.73 114.96 186.98  307.80  2004 1.00
## y_rep[307]     295.25    2.75 118.55 167.70 273.88  448.88  1859 1.00
## y_rep[308]     226.21    1.98  91.00 125.57 211.86  341.98  2116 1.00
## y_rep[309]     307.42    2.90 126.27 173.94 285.16  472.36  1890 1.00
## y_rep[310]     451.19    4.96 186.29 249.41 420.33  687.48  1411 1.00
## y_rep[311]     196.21    1.81  83.82 110.67 179.73  298.44  2140 1.00
## y_rep[312]     308.03    2.84 126.91 172.37 285.31  469.91  2000 1.00
## y_rep[313]     225.46    2.24  96.05 122.80 207.43  346.62  1831 1.00
## y_rep[314]     452.02    4.23 189.40 254.33 414.34  697.38  2007 1.00
## y_rep[315]     206.29    1.79  82.84 117.46 191.16  314.46  2154 1.00
## y_rep[316]     197.87    1.82  81.08 109.39 182.38  302.23  1983 1.00
## y_rep[317]     441.50    4.04 183.17 247.83 403.89  675.75  2058 1.00
## y_rep[318]     204.05    2.02  85.98 112.13 187.85  316.27  1813 1.00
## y_rep[319]     300.83    2.75 124.64 171.92 276.86  459.02  2052 1.00
## y_rep[320]     291.72    2.70 118.63 160.35 269.62  445.20  1925 1.00
## y_rep[321]     636.39    6.16 272.05 345.79 583.99  982.35  1949 1.00
## y_rep[322]     603.14    5.85 260.32 335.49 552.64  930.41  1983 1.00
## y_rep[323]     205.04    2.02  83.85 113.88 189.76  314.93  1726 1.00
## y_rep[324]     689.86    6.65 280.07 383.75 640.67 1052.79  1772 1.00
## y_rep[325]     206.50    1.88  84.90 114.65 191.45  313.72  2044 1.00
## y_rep[326]     225.77    2.17  93.30 124.01 208.64  347.49  1850 1.00
## y_rep[327]     290.36    2.69 121.28 159.91 264.33  454.58  2030 1.00
## y_rep[328]     307.09    2.80 124.06 173.47 285.97  475.21  1960 1.00
## y_rep[329]     306.07    2.98 131.65 166.95 284.03  471.61  1956 1.00
## y_rep[330]     445.65    4.33 182.91 249.02 413.83  685.10  1781 1.00
## y_rep[331]     619.94    5.61 256.49 334.94 573.46  949.76  2090 1.00
## y_rep[332]     195.25    1.81  80.56 104.37 179.71  300.02  1985 1.00
## y_rep[333]     346.18    3.40 154.71 185.94 319.04  535.44  2070 1.00
## y_rep[334]     499.13    4.91 216.54 275.13 454.07  775.92  1945 1.00
## y_rep[335]     499.73    4.65 206.29 280.58 463.26  755.81  1969 1.00
## y_rep[336]     630.77    6.19 269.73 343.84 585.38  976.58  1902 1.00
## y_rep[337]     710.35    6.52 291.49 385.75 659.87 1095.89  2001 1.00
## y_rep[338]     205.52    1.91  84.00 116.74 189.21  315.11  1928 1.00
## y_rep[339]     312.14    2.96 133.08 172.48 287.92  486.31  2024 1.00
## y_rep[340]     196.07    1.89  81.80 109.54 182.08  297.47  1869 1.00
## y_rep[341]     201.41    1.86  84.56 111.22 185.36  309.35  2061 1.00
## y_rep[342]     228.24    2.10  95.10 127.03 209.95  350.65  2048 1.00
## y_rep[343]     343.63    3.20 142.24 190.14 322.15  525.20  1978 1.00
## y_rep[344]     455.88    4.09 183.95 255.20 422.23  695.52  2020 1.00
## y_rep[345]     196.57    1.82  82.76 108.16 181.06  303.13  2062 1.00
## y_rep[346]     299.21    2.89 127.59 164.80 273.67  460.96  1943 1.00
## y_rep[347]     199.97    1.82  79.03 112.43 187.10  303.61  1886 1.00
## y_rep[348]     305.84    2.89 125.08 167.98 283.22  475.23  1872 1.00
## y_rep[349]     635.17    6.16 266.93 348.97 580.65  976.05  1879 1.00
## y_rep[350]     293.65    2.73 122.59 163.28 270.83  448.94  2023 1.00
## y_rep[351]     456.17    4.28 184.76 249.82 426.06  695.84  1864 1.00
## y_rep[352]     497.35    4.79 206.55 278.03 459.21  769.48  1859 1.00
## y_rep[353]     342.32    3.25 143.40 189.66 314.46  539.07  1952 1.00
## y_rep[354]     201.40    1.87  84.65 112.14 187.78  303.62  2050 1.00
## y_rep[355]     702.59    6.83 292.81 390.05 648.91 1081.09  1836 1.00
## y_rep[356]     618.94    5.71 250.95 340.91 573.88  956.58  1931 1.00
## y_rep[357]     227.39    2.17  96.31 123.14 208.45  356.41  1966 1.00
## y_rep[358]     451.06    4.40 193.13 243.52 414.66  698.51  1924 1.00
## y_rep[359]     632.64    5.79 256.55 356.67 592.82  951.82  1966 1.00
## y_rep[360]     445.02    4.31 189.71 243.98 406.30  687.20  1937 1.00
## y_rep[361]     224.66    2.14  91.58 127.77 208.22  351.12  1829 1.00
## y_rep[362]     506.89    4.80 213.35 281.95 467.89  781.22  1974 1.00
## y_rep[363]     307.39    3.14 128.92 170.53 281.80  474.40  1691 1.00
## y_rep[364]     203.11    1.83  80.90 113.26 189.58  318.71  1959 1.00
## y_rep[365]     290.17    2.63 121.18 165.24 272.10  439.40  2122 1.00
## y_rep[366]     202.55    1.96  84.64 113.17 187.55  314.58  1862 1.00
## y_rep[367]     695.00    6.84 303.60 378.02 640.69 1075.40  1971 1.00
## y_rep[368]     626.80    5.73 259.09 348.42 580.49  954.11  2043 1.00
## y_rep[369]     196.61    1.81  79.45 111.75 181.80  299.03  1934 1.00
## y_rep[370]     342.16    3.44 150.60 187.67 313.78  537.02  1916 1.00
## y_rep[371]     450.70    4.03 179.03 254.28 421.25  684.97  1973 1.00
## y_rep[372]     305.99    2.83 125.75 170.69 286.21  467.94  1976 1.00
## y_rep[373]     207.78    1.94  86.98 113.23 191.18  323.11  2019 1.00
## y_rep[374]     628.90    5.87 257.13 357.00 584.07  952.95  1919 1.00
## y_rep[375]     604.20    5.54 251.48 336.74 557.28  949.72  2063 1.00
## y_rep[376]     205.26    1.95  85.12 112.62 189.70  315.79  1910 1.00
## y_rep[377]     684.03    6.48 280.93 388.42 621.79 1064.53  1879 1.00
## y_rep[378]     196.82    1.82  82.65 108.95 181.82  301.74  2053 1.00
## y_rep[379]     461.93    4.51 194.54 256.27 423.74  721.53  1859 1.00
## y_rep[380]     295.22    2.84 124.84 161.01 269.67  450.92  1935 1.00
## y_rep[381]     311.49    2.89 131.18 170.14 287.80  483.20  2061 1.00
## y_rep[382]     503.17    5.02 211.24 284.04 467.01  759.20  1767 1.00
## y_rep[383]     439.86    4.13 185.33 241.98 407.32  674.42  2018 1.00
## y_rep[384]     336.47    3.16 140.42 186.19 310.29  524.49  1981 1.00
## y_rep[385]     227.53    2.15  94.53 125.14 210.80  352.95  1941 1.00
## y_rep[386]     301.60    3.06 124.98 168.35 277.71  463.98  1668 1.00
## y_rep[387]     293.46    2.93 123.06 159.66 270.67  456.53  1761 1.00
## y_rep[388]     228.85    2.11  97.87 125.28 213.04  354.90  2147 1.00
## y_rep[389]     603.68    5.73 247.47 344.41 552.48  919.73  1864 1.00
## y_rep[390]     619.90    5.76 252.41 342.25 578.90  948.47  1919 1.00
## y_rep[391]     494.66    4.69 200.37 273.98 461.17  756.21  1826 1.00
## y_rep[392]     307.25    2.80 124.69 171.13 282.38  463.91  1985 1.00
## y_rep[393]     452.54    4.42 190.23 244.67 419.53  703.76  1853 1.00
## y_rep[394]     441.63    4.07 186.80 244.31 407.59  667.64  2102 1.00
## y_rep[395]     203.78    1.95  85.92 112.23 188.98  316.34  1940 1.00
## y_rep[396]     196.62    1.94  82.97 109.05 180.88  301.58  1831 1.00
## y_rep[397]     205.09    1.96  85.82 114.09 190.31  314.07  1912 1.00
## y_rep[398]     346.85    3.49 142.29 186.70 326.76  534.88  1664 1.00
## y_rep[399]     225.69    2.11  93.35 124.98 210.41  344.88  1954 1.00
## y_rep[400]     194.84    1.80  78.83 109.38 180.81  300.10  1912 1.00
## y_rep[401]     699.35    6.60 293.95 383.99 652.38 1071.51  1984 1.00
## y_rep[402]     200.89    2.10  85.07 109.22 185.59  306.04  1636 1.00
## y_rep[403]     496.86    4.58 207.14 276.24 459.85  764.37  2046 1.00
## y_rep[404]     205.67    1.93  85.75 113.28 190.77  313.25  1971 1.00
## y_rep[405]     439.53    4.13 182.55 242.06 408.26  665.88  1958 1.00
## y_rep[406]     593.61    5.37 240.93 330.21 550.30  908.19  2013 1.00
## y_rep[407]     336.88    3.20 139.06 188.93 311.48  505.76  1892 1.00
## y_rep[408]     297.05    2.84 125.84 164.12 272.63  452.45  1969 1.00
## y_rep[409]     302.54    2.76 121.82 169.15 281.91  463.44  1950 1.00
## y_rep[410]     308.62    3.07 129.92 171.26 284.36  470.01  1796 1.00
## y_rep[411]     197.01    1.75  78.49 110.84 182.93  299.60  2002 1.00
## y_rep[412]     618.34    5.74 248.92 347.87 574.09  951.25  1878 1.00
## y_rep[413]     304.21    2.74 125.27 169.59 279.47  469.50  2083 1.00
## y_rep[414]     451.53    4.33 186.45 247.54 417.82  698.51  1852 1.00
## y_rep[415]     224.89    2.17  92.59 124.56 209.09  344.92  1826 1.00
## y_rep[416]     335.46    3.40 136.78 185.64 309.52  514.13  1622 1.00
## y_rep[417]     292.31    2.69 119.46 161.52 269.70  447.14  1974 1.00
## y_rep[418]     630.25    6.03 261.61 362.62 585.00  949.28  1884 1.00
## y_rep[419]     202.95    1.95  84.53 110.98 189.56  311.45  1883 1.00
## y_rep[420]     457.41    4.11 187.29 254.41 425.65  696.91  2076 1.00
## y_rep[421]     203.76    2.01  83.22 112.50 189.94  316.49  1712 1.00
## y_rep[422]     342.27    3.34 145.17 188.89 318.06  519.56  1894 1.00
## y_rep[423]     453.42    4.17 184.80 252.79 421.44  699.70  1963 1.00
## y_rep[424]     308.06    2.98 128.89 172.82 283.29  479.73  1871 1.00
## y_rep[425]     291.08    2.63 116.78 164.03 270.27  450.47  1970 1.00
## y_rep[426]     208.79    2.05  88.73 114.01 192.72  320.86  1873 1.00
## y_rep[427]     501.20    4.70 205.67 281.61 468.88  757.66  1913 1.00
## y_rep[428]     304.15    2.67 125.15 171.81 280.25  469.36  2198 1.00
## y_rep[429]     225.73    2.17  92.55 127.12 206.78  348.46  1820 1.00
## y_rep[430]     696.85    5.93 283.30 388.66 647.90 1061.50  2280 1.00
## y_rep[431]     195.85    1.78  81.35 111.28 181.65  304.19  2088 1.00
## y_rep[432]     451.39    4.33 193.31 251.54 415.62  688.43  1991 1.00
## y_rep[433]     626.80    5.97 258.28 347.62 580.78  965.33  1872 1.00
## y_rep[434]     204.28    1.98  87.69 111.43 188.66  313.01  1963 1.00
## y_rep[435]     620.92    5.84 256.81 340.43 574.30  957.84  1931 1.00
## y_rep[436]     295.45    2.90 123.98 166.07 270.64  458.13  1830 1.00
## y_rep[437]     500.53    4.88 214.39 272.33 457.30  781.60  1933 1.00
## y_rep[438]     197.29    1.87  82.22 108.36 183.66  298.06  1935 1.00
## y_rep[439]     228.13    2.15  94.51 126.34 211.05  353.01  1924 1.00
## y_rep[440]     201.88    1.91  84.55 111.81 185.69  312.09  1969 1.00
## y_rep[441]     691.58    6.48 284.43 386.31 638.50 1058.11  1926 1.00
## y_rep[442]     343.23    3.32 150.51 190.27 312.21  528.12  2057 1.00
## y_rep[443]     206.03    1.81  81.97 113.31 190.89  311.30  2048 1.00
## y_rep[444]     228.02    2.15  93.43 125.31 210.64  357.25  1887 1.00
## y_rep[445]     605.63    5.53 252.69 332.73 561.97  945.38  2087 1.00
## y_rep[446]     628.06    6.04 262.92 351.94 573.78  958.36  1895 1.00
## y_rep[447]     429.34    3.77 174.84 235.82 401.20  656.91  2152 1.00
## y_rep[448]     688.19    6.55 282.50 378.82 638.97 1036.67  1862 1.00
## y_rep[449]     500.72    4.69 206.41 282.09 462.99  760.45  1937 1.00
## y_rep[450]     291.62    2.69 117.88 165.79 270.76  444.90  1924 1.00
## y_rep[451]     305.06    2.85 124.96 170.32 283.68  466.81  1927 1.00
## y_rep[452]     457.55    4.40 193.35 255.63 419.09  710.47  1930 1.00
## y_rep[453]     207.88    1.86  84.15 117.33 192.19  319.60  2048 1.00
## y_rep[454]     300.63    2.86 127.47 163.12 273.63  469.81  1984 1.00
## y_rep[455]     202.41    1.89  84.08 110.30 188.54  308.44  1972 1.00
## y_rep[456]     199.11    1.97  80.35 110.23 184.35  307.64  1667 1.00
## y_rep[457]     334.28    2.94 134.94 186.35 308.48  513.79  2100 1.00
## y_rep[458]     446.62    4.21 186.34 251.22 414.62  670.14  1964 1.00
## y_rep[459]     198.10    2.03  81.42 107.60 184.99  307.85  1610 1.00
## y_rep[460]     207.86    2.03  88.76 113.52 193.05  320.89  1909 1.00
## y_rep[461]     298.88    2.78 125.90 163.76 275.95  458.30  2056 1.00
## y_rep[462]     225.45    2.09  90.67 126.56 208.73  343.43  1877 1.00
## y_rep[463]     203.86    1.86  85.19 112.83 186.78  319.33  2107 1.00
## y_rep[464]     304.90    2.84 125.65 171.32 282.61  465.55  1953 1.00
## y_rep[465]     203.43    1.96  87.36 109.32 186.98  318.28  1994 1.00
## y_rep[466]     195.53    1.81  83.67 109.59 181.97  298.80  2128 1.00
## y_rep[467]     228.04    2.12  93.41 124.95 211.77  353.02  1939 1.00
## y_rep[468]     500.95    4.62 207.83 276.31 463.06  771.19  2025 1.00
## y_rep[469]     311.16    2.86 132.13 168.35 285.18  485.54  2127 1.00
## y_rep[470]     298.49    2.86 124.83 166.22 273.57  463.36  1900 1.00
## y_rep[471]     342.27    3.25 144.82 186.98 316.56  531.90  1985 1.00
## y_rep[472]     697.38    6.64 292.14 385.69 640.43 1084.23  1938 1.00
## y_rep[473]     638.88    6.03 260.88 355.56 590.69  987.66  1873 1.00
## y_rep[474]     455.39    4.13 190.03 249.89 422.55  702.02  2119 1.00
## y_rep[475]     206.39    1.90  86.87 113.95 189.02  321.89  2088 1.00
## y_rep[476]     451.51    4.21 184.23 252.75 416.71  687.10  1917 1.00
## y_rep[477]     623.15    5.84 267.61 345.55 568.82  975.82  2101 1.00
## y_rep[478]     694.52    6.42 290.45 382.36 637.39 1089.38  2049 1.00
## y_rep[479]     208.18    2.11  89.96 115.12 190.94  318.73  1814 1.00
## y_rep[480]     341.50    3.10 140.24 189.81 314.03  530.49  2050 1.00
## y_rep[481]     228.48    2.09  94.87 127.43 211.57  354.23  2055 1.00
## y_rep[482]     205.72    1.92  86.10 115.50 190.13  314.78  2020 1.00
## y_rep[483]     195.79    1.80  81.56 106.82 181.30  302.33  2047 1.00
## y_rep[484]     306.38    2.72 124.66 171.42 284.00  466.83  2097 1.00
## y_rep[485]     495.49    4.55 202.85 277.49 461.44  754.35  1989 1.00
## y_rep[486]     306.17    3.20 131.10 168.48 283.28  466.38  1681 1.00
## y_rep[487]     629.22    6.10 268.79 343.10 581.11  974.60  1944 1.00
## y_rep[488]     225.72    2.32  97.08 125.34 204.89  349.78  1747 1.00
## y_rep[489]     451.78    4.43 185.08 249.71 418.55  694.21  1748 1.00
## y_rep[490]     339.04    3.09 138.35 184.80 314.96  526.05  2002 1.00
## y_rep[491]     452.07    4.16 181.01 257.91 416.54  687.35  1897 1.00
## y_rep[492]     494.61    4.67 204.91 274.28 451.87  759.98  1926 1.00
## y_rep[493]     203.24    1.95  84.95 111.94 188.82  313.30  1902 1.00
## y_rep[494]     194.02    1.79  79.01 108.58 179.55  295.15  1946 1.00
## y_rep[495]     291.80    2.74 123.28 162.99 266.56  454.53  2024 1.00
## y_rep[496]     694.53    6.56 287.98 384.19 643.80 1068.25  1926 1.00
## y_rep[497]     203.70    1.99  85.81 113.03 189.64  312.73  1864 1.00
## y_rep[498]     621.09    5.60 251.98 347.04 569.58  956.75  2028 1.00
## y_rep[499]     308.26    3.00 128.79 168.23 285.40  476.19  1848 1.00
## y_rep[500]     294.73    2.85 124.14 162.05 276.06  455.11  1900 1.00
## y_rep[501]     204.40    2.02  86.78 116.59 184.83  315.94  1851 1.00
## y_rep[502]     696.65    6.87 292.78 381.49 643.53 1070.84  1818 1.00
## y_rep[503]     498.59    4.68 211.95 283.87 449.96  770.03  2048 1.00
## y_rep[504]     452.21    4.44 184.88 252.21 419.61  686.24  1737 1.00
## y_rep[505]     621.16    5.81 254.87 340.24 571.51  965.64  1923 1.00
## y_rep[506]     202.74    1.94  83.81 112.76 186.26  311.55  1867 1.00
## y_rep[507]     339.38    3.21 140.60 189.64 313.57  522.20  1923 1.00
## y_rep[508]     226.78    2.14  94.71 126.81 208.60  347.09  1953 1.00
## y_rep[509]     308.81    2.89 126.17 169.75 283.53  483.66  1902 1.00
## y_rep[510]     304.73    2.76 123.13 170.18 284.28  466.19  1983 1.00
## y_rep[511]     194.77    1.76  78.64 109.64 181.65  297.39  2008 1.00
## y_rep[512]     635.53    5.93 261.79 351.65 589.13  985.94  1952 1.00
## y_rep[513]     207.48    1.87  84.08 116.69 190.28  316.67  2014 1.00
## y_rep[514]     314.15    2.93 131.41 172.79 290.25  484.19  2012 1.00
## y_rep[515]     283.08    2.42 111.36 156.82 264.25  434.55  2115 1.00
## y_rep[516]     585.49    5.48 238.33 327.36 548.05  903.87  1892 1.00
## y_rep[517]     179.64    1.73  73.75  99.19 165.61  280.70  1821 1.00
## y_rep[518]     424.46    3.90 173.84 233.90 390.38  656.80  1982 1.00
## y_rep[519]     185.73    1.86  81.69 102.62 167.82  292.75  1932 1.00
## y_rep[520]     419.15    3.80 176.22 229.46 384.51  648.67  2150 1.00
## y_rep[521]     188.20    1.91  81.43 103.65 171.54  291.85  1822 1.00
## y_rep[522]     598.30    5.69 247.66 326.01 552.16  931.65  1897 1.00
## y_rep[523]     277.96    2.69 113.05 156.28 254.93  423.04  1761 1.00
## y_rep[524]     644.27    6.20 270.84 348.45 597.33 1000.34  1909 1.00
## y_rep[525]     465.42    4.27 198.00 260.56 427.43  706.47  2147 1.00
## y_rep[526]     593.60    5.66 248.72 323.54 546.40  918.45  1933 1.00
## y_rep[527]     460.25    4.30 187.65 258.08 428.18  707.18  1907 1.00
## y_rep[528]     573.28    5.86 249.33 310.67 525.56  882.78  1811 1.00
## y_rep[529]     270.06    2.55 113.41 150.16 249.14  415.30  1973 1.00
## y_rep[530]     280.83    2.53 114.76 155.04 262.41  442.70  2055 1.00
## y_rep[531]     184.94    1.77  78.02 102.91 168.10  282.19  1939 1.00
## y_rep[532]     400.66    3.84 171.51 216.00 371.18  620.23  1996 1.00
## y_rep[533]     416.84    4.14 173.86 234.06 383.66  642.27  1764 1.00
## y_rep[534]     209.95    2.02  87.69 118.04 192.67  326.26  1887 1.00
## y_rep[535]     285.26    2.70 119.05 159.19 265.50  436.98  1938 1.00
## y_rep[536]     644.76    6.36 268.56 356.69 589.79 1015.84  1785 1.00
## y_rep[537]     181.84    1.67  74.34 102.18 168.43  276.80  1978 1.00
## y_rep[538]     311.63    3.01 129.09 173.69 288.48  476.17  1838 1.00
## y_rep[539]     188.39    1.79  78.69 103.67 173.21  292.46  1932 1.00
## y_rep[540]     464.92    4.31 195.36 257.71 425.97  717.07  2051 1.00
## y_rep[541]     403.15    3.61 165.69 225.53 372.50  620.64  2109 1.00
## y_rep[542]     564.60    5.16 238.48 309.89 520.16  868.82  2134 1.00
## y_rep[543]     427.33    4.19 182.28 233.77 393.48  662.70  1888 1.00
## y_rep[544]     279.72    2.67 116.77 155.05 256.00  433.31  1912 1.00
## y_rep[545]     185.71    1.77  77.23 100.58 172.25  282.66  1912 1.00
## y_rep[546]     181.33    1.79  78.35  97.73 167.17  286.44  1914 1.00
## y_rep[547]     283.98    2.64 119.44 155.45 263.76  437.43  2041 1.00
## y_rep[548]     418.70    3.93 173.07 230.95 390.36  636.75  1943 1.00
## y_rep[549]     267.74    2.50 109.84 148.74 247.31  412.04  1935 1.00
## y_rep[550]     602.45    5.74 258.17 325.05 559.72  928.98  2021 1.00
## y_rep[551]     586.59    5.47 245.32 327.25 535.83  898.08  2012 1.00
## y_rep[552]     185.40    1.71  75.95 103.67 172.18  286.37  1974 1.00
## y_rep[553]     663.91    6.48 282.52 360.97 614.50 1023.20  1903 1.01
## y_rep[554]     314.07    2.92 127.95 173.73 292.80  477.78  1921 1.00
## y_rep[555]     208.78    2.11  88.41 116.37 194.49  321.11  1751 1.00
## y_rep[556]     281.16    2.59 115.51 157.17 260.41  434.27  1988 1.00
## y_rep[557]     186.52    1.77  77.01 100.61 173.65  291.46  1891 1.00
## y_rep[558]     400.86    4.06 172.91 222.51 363.02  618.23  1811 1.00
## y_rep[559]     205.97    1.92  87.58 112.36 191.22  316.55  2083 1.00
## y_rep[560]     176.75    1.68  72.62  97.83 162.61  273.38  1874 1.00
## y_rep[561]     570.66    5.55 245.03 311.78 518.59  890.16  1946 1.00
## y_rep[562]     589.57    5.04 229.95 318.86 550.96  899.43  2081 1.00
## y_rep[563]     319.61    2.93 135.04 176.01 295.32  491.52  2129 1.00
## y_rep[564]     416.00    3.74 164.92 234.99 386.07  631.91  1948 1.00
## y_rep[565]     269.64    2.58 112.26 148.43 249.69  411.97  1896 1.00
## y_rep[566]     187.33    1.82  79.06 100.07 171.37  293.49  1897 1.00
## y_rep[567]     281.99    2.76 122.07 154.47 259.09  426.93  1950 1.00
## y_rep[568]     186.66    1.66  75.25 104.17 172.54  285.45  2046 1.00
## y_rep[569]     569.99    5.33 237.32 314.82 524.87  870.22  1986 1.00
## y_rep[570]     280.99    2.77 114.16 158.01 262.30  423.54  1695 1.00
## y_rep[571]     423.61    4.03 181.92 229.87 385.50  666.76  2040 1.00
## y_rep[572]     594.64    5.88 246.83 328.46 545.63  903.41  1765 1.00
## y_rep[573]     281.11    2.68 118.15 157.77 258.42  439.50  1950 1.00
## y_rep[574]     422.17    3.90 180.47 226.81 387.25  655.77  2142 1.00
## y_rep[575]     463.92    4.18 189.20 256.27 427.88  706.87  2053 1.00
## y_rep[576]     205.46    1.84  85.09 113.47 189.17  324.11  2150 1.00
## y_rep[577]     270.39    2.58 115.08 147.36 251.42  418.85  1990 1.00
## y_rep[578]     309.08    2.81 127.29 171.25 288.13  473.56  2049 1.00
## y_rep[579]     649.25    6.30 274.79 354.75 606.76 1022.17  1904 1.00
## y_rep[580]     183.10    1.78  78.43  97.41 169.16  285.78  1946 1.00
## y_rep[581]     579.90    5.79 247.08 320.37 533.51  887.80  1824 1.00
## y_rep[582]     402.50    3.76 167.24 219.59 375.15  613.71  1983 1.00
## y_rep[583]     188.96    1.86  79.96 102.85 174.41  292.70  1852 1.00
## y_rep[584]     183.80    1.69  74.46 102.67 171.60  279.20  1935 1.00
## y_rep[585]     277.28    2.50 115.08 151.03 257.55  427.20  2115 1.00
## y_rep[586]     207.66    1.92  85.77 113.31 193.84  315.29  2001 1.00
## y_rep[587]     468.71    4.24 191.03 262.04 430.10  720.96  2029 1.00
## y_rep[588]     317.61    3.41 134.19 175.88 291.01  492.79  1553 1.00
## y_rep[589]     414.97    3.80 171.54 230.25 389.86  640.56  2036 1.00
## y_rep[590]     267.25    2.51 112.93 146.76 246.29  409.35  2029 1.00
## y_rep[591]     588.86    5.82 250.11 329.12 539.81  898.28  1850 1.00
## y_rep[592]     652.57    6.29 267.11 360.39 602.49  992.51  1805 1.00
## y_rep[593]     282.13    2.83 118.70 154.05 261.77  438.62  1762 1.00
## y_rep[594]     424.27    3.95 170.73 237.01 395.49  648.89  1868 1.00
## y_rep[595]     181.08    1.77  75.45 100.98 167.17  275.71  1810 1.00
## y_rep[596]     184.74    1.75  75.31 101.78 169.57  288.93  1845 1.00
## y_rep[597]     579.56    5.23 234.80 318.27 536.66  889.88  2015 1.00
## y_rep[598]     568.02    5.89 236.76 317.02 523.91  878.74  1616 1.00
## y_rep[599]     463.51    4.21 190.53 255.63 433.12  701.38  2051 1.00
## y_rep[600]     203.24    1.89  85.34 110.72 188.93  312.48  2049 1.00
## y_rep[601]     595.44    6.07 265.49 327.68 539.39  934.41  1912 1.00
## y_rep[602]     279.79    2.70 112.89 155.84 260.39  428.54  1744 1.00
## y_rep[603]     185.63    1.63  76.09 100.58 173.12  286.96  2177 1.00
## y_rep[604]     272.78    2.63 116.82 149.36 248.68  431.10  1973 1.00
## y_rep[605]     424.85    4.06 177.80 229.64 393.33  654.03  1922 1.00
## y_rep[606]     655.14    6.38 287.43 362.64 598.31 1016.81  2028 1.00
## y_rep[607]     177.10    1.79  74.06  99.76 162.61  278.10  1720 1.00
## y_rep[608]     398.30    3.88 165.82 222.08 369.06  607.32  1822 1.00
## y_rep[609]     184.61    1.80  77.94 103.08 170.79  280.65  1874 1.00
## y_rep[610]     304.26    2.96 125.27 171.78 282.10  468.69  1797 1.00
## y_rep[611]     676.23    6.14 276.47 377.75 623.03 1041.94  2029 1.00
## y_rep[612]     273.26    2.57 112.28 151.36 255.90  417.53  1906 1.00
## y_rep[613]     206.13    2.02  83.04 116.56 191.76  318.35  1694 1.01
## y_rep[614]     577.56    5.86 242.84 325.43 530.55  884.05  1719 1.00
## y_rep[615]     424.02    4.11 181.49 231.62 390.62  652.14  1949 1.00
## y_rep[616]     421.15    4.05 176.45 232.20 389.45  651.03  1894 1.00
## y_rep[617]     283.85    2.64 119.29 154.05 262.91  441.46  2046 1.00
## y_rep[618]     316.23    2.88 134.42 176.63 287.03  488.86  2173 1.00
## y_rep[619]     598.22    5.72 257.23 327.82 549.50  931.13  2025 1.00
## y_rep[620]     181.70    1.72  74.03 102.73 166.44  280.65  1855 1.00
## y_rep[621]     186.68    1.77  79.98 102.66 171.88  295.34  2037 1.00
## y_rep[622]     469.55    4.85 203.09 251.69 429.99  716.27  1751 1.00
## y_rep[623]     187.21    1.87  76.61 101.36 174.06  292.04  1686 1.00
## y_rep[624]     315.15    3.00 134.40 173.57 287.26  490.70  2008 1.00
## y_rep[625]     646.02    5.81 263.64 361.86 600.39  979.36  2062 1.00
## y_rep[626]     192.09    1.80  80.69 107.19 175.35  301.66  2007 1.00
## y_rep[627]     280.64    2.67 118.27 153.73 260.66  442.53  1968 1.00
## y_rep[628]     593.46    5.77 249.37 328.90 545.57  905.07  1867 1.00
## y_rep[629]     267.54    2.51 111.49 151.51 245.63  412.19  1968 1.00
## y_rep[630]     179.10    1.74  73.23  99.73 165.66  274.21  1771 1.00
## y_rep[631]     416.07    3.83 167.89 229.69 392.74  637.93  1922 1.00
## y_rep[632]     462.97    4.35 191.82 255.63 429.92  708.56  1940 1.00
## y_rep[633]     284.74    2.51 116.75 159.90 261.46  444.45  2161 1.00
## y_rep[634]     205.82    1.87  85.63 113.51 189.87  313.66  2105 1.00
## y_rep[635]     183.99    1.65  74.01 102.16 169.25  287.24  2004 1.00
## y_rep[636]     202.73    1.84  82.41 113.00 187.48  311.39  2004 1.00
## y_rep[637]     202.06    1.92  84.67 111.07 186.03  310.98  1936 1.00
## y_rep[638]     620.23    5.86 260.28 341.79 566.92  958.88  1975 1.00
## y_rep[639]     432.57    4.09 173.48 241.66 405.69  654.76  1801 1.00
## y_rep[640]     300.47    2.86 125.91 168.96 277.77  463.06  1940 1.00
## y_rep[641]     291.46    2.62 118.10 159.74 269.62  451.67  2038 1.00
## y_rep[642]     195.54    1.79  80.19 107.58 181.29  297.93  1996 1.00
## y_rep[643]     305.44    2.86 126.92 169.32 283.28  482.26  1975 1.00
## y_rep[644]     290.40    2.81 123.47 160.95 265.37  451.52  1932 1.00
## y_rep[645]     433.92    4.07 178.18 241.02 406.00  668.90  1916 1.00
## y_rep[646]     626.26    6.17 265.75 339.79 574.46  976.22  1858 1.00
## y_rep[647]     446.79    4.16 183.19 249.05 417.15  686.22  1942 1.00
## y_rep[648]     646.05    5.99 269.03 362.05 595.74 1003.52  2014 1.00
## y_rep[649]     298.55    2.67 118.60 167.30 282.45  458.76  1975 1.00
## y_rep[650]     649.67    6.24 269.21 359.79 603.32  992.21  1861 1.00
## y_rep[651]     725.53    6.89 309.11 397.06 666.68 1122.82  2013 1.00
## y_rep[652]     645.32    5.94 273.01 356.47 598.61  984.77  2114 1.00
## y_rep[653]     438.59    4.21 185.05 243.74 405.13  676.76  1933 1.00
## y_rep[654]     453.59    4.41 194.30 245.04 418.46  711.60  1944 1.00
## y_rep[655]     291.79    2.63 117.83 163.65 270.60  448.84  2005 1.00
## y_rep[656]     617.02    5.90 256.44 336.54 575.53  934.75  1892 1.00
## y_rep[657]     340.30    3.08 140.76 187.82 313.92  526.23  2088 1.00
## y_rep[658]     307.10    3.10 134.49 170.32 280.13  475.31  1887 1.00
## y_rep[659]     509.93    4.61 212.90 280.84 464.67  796.16  2133 1.00
## y_rep[660]     453.23    4.23 189.67 250.32 420.03  696.19  2011 1.00
## y_rep[661]     445.23    3.98 181.16 249.95 409.48  681.90  2076 1.00
## y_rep[662]     513.03    4.82 214.36 281.37 478.36  779.63  1975 1.00
## y_rep[663]     340.52    3.49 144.23 188.51 313.98  529.91  1704 1.00
## y_rep[664]     295.26    2.68 122.48 168.02 270.91  449.12  2090 1.00
## y_rep[665]     715.23    7.10 301.67 397.42 661.88 1087.34  1807 1.00
## y_rep[666]     626.00    5.67 258.71 345.98 572.32  961.35  2078 1.00
## y_rep[667]     194.94    1.75  78.48 107.89 179.42  301.01  2014 1.00
## y_rep[668]     305.10    2.78 124.96 170.89 282.29  463.91  2027 1.00
## y_rep[669]     658.43    6.51 276.09 363.36 607.02 1022.08  1799 1.00
## y_rep[670]     339.69    3.53 143.88 186.30 315.02  518.96  1662 1.00
## y_rep[671]     197.90    1.70  79.40 111.64 181.60  300.77  2172 1.00
## y_rep[672]     505.00    4.80 209.79 282.68 462.06  777.99  1914 1.00
## y_rep[673]     224.03    2.26  94.63 123.64 205.79  349.96  1758 1.00
## y_rep[674]     721.67    7.00 303.14 404.89 667.99 1111.46  1874 1.00
## y_rep[675]     460.14    4.40 195.44 256.75 426.30  696.63  1977 1.00
## y_rep[676]     203.46    1.93  85.84 111.76 186.17  318.18  1975 1.00
## y_rep[677]     199.32    1.93  85.79 106.86 183.25  311.39  1965 1.00
## y_rep[678]     506.90    4.63 203.26 277.59 477.43  773.66  1928 1.00
## y_rep[679]     307.79    2.87 132.59 169.44 281.31  477.66  2137 1.00
## y_rep[680]     200.89    1.92  84.30 109.53 186.40  309.39  1918 1.00
## y_rep[681]     652.48    6.26 269.28 366.59 603.43  991.02  1851 1.00
## y_rep[682]     456.61    4.46 190.50 251.91 422.11  702.39  1827 1.00
## y_rep[683]     226.65    2.16  96.95 121.93 209.89  348.68  2020 1.00
## y_rep[684]     192.89    1.71  77.59 108.05 179.30  291.49  2066 1.00
## y_rep[685]     333.50    3.13 136.57 186.01 308.57  520.21  1908 1.00
## y_rep[686]     731.08    6.71 302.70 401.05 677.40 1115.32  2033 1.00
## y_rep[687]     192.25    1.77  79.24 107.14 176.89  296.06  1997 1.00
## y_rep[688]     660.07    6.33 285.83 363.55 610.06 1021.44  2042 1.00
## y_rep[689]     306.86    2.94 128.45 171.34 283.63  464.59  1912 1.00
## y_rep[690]     224.86    2.05  94.27 123.28 208.00  345.43  2122 1.00
## y_rep[691]     203.26    1.96  88.17 112.52 186.32  313.03  2016 1.00
## y_rep[692]     290.49    2.97 123.33 161.18 266.80  445.89  1729 1.00
## y_rep[693]     201.21    1.88  89.06 107.54 183.80  322.88  2241 1.00
## y_rep[694]     460.22    4.05 184.39 257.78 424.41  707.56  2075 1.00
## y_rep[695]     300.72    2.68 122.33 166.59 280.87  462.86  2076 1.00
## y_rep[696]     651.57    6.04 269.58 356.63 605.66 1003.17  1995 1.00
## y_rep[697]     194.12    1.89  81.89 103.94 178.69  299.82  1882 1.00
## y_rep[698]     225.49    2.10  95.23 127.51 206.20  339.84  2048 1.00
## y_rep[699]     292.85    2.79 120.38 162.29 270.78  451.51  1856 1.00
## y_rep[700]     459.39    4.50 199.21 249.32 420.64  722.19  1964 1.00
## y_rep[701]     309.63    2.90 126.14 169.73 289.36  472.35  1893 1.00
## y_rep[702]     206.00    1.94  87.17 113.58 188.96  318.63  2017 1.00
## y_rep[703]     197.91    1.97  86.44 107.11 183.88  298.87  1934 1.00
## y_rep[704]     337.23    2.99 142.24 188.14 307.09  523.76  2266 1.00
## y_rep[705]     341.14    3.32 146.20 186.65 314.62  532.59  1934 1.00
## y_rep[706]     303.07    2.87 126.99 167.25 282.13  467.56  1960 1.00
## y_rep[707]     196.98    1.78  81.16 110.14 182.32  299.71  2073 1.00
## y_rep[708]     192.55    1.82  79.46 108.92 176.12  299.86  1916 1.00
## y_rep[709]     200.32    1.97  87.13 110.33 183.32  311.09  1948 1.00
## y_rep[710]     642.81    6.09 263.49 357.79 596.80  980.21  1870 1.00
## y_rep[711]     446.25    4.20 185.76 248.31 408.89  688.50  1956 1.00
## y_rep[712]     720.80    6.84 315.51 395.04 668.14 1108.81  2130 1.00
## y_rep[713]     221.19    2.09  92.24 122.59 202.09  338.85  1941 1.00
## y_rep[714]     289.41    2.64 116.75 159.18 270.13  440.49  1959 1.00
## y_rep[715]     502.24    4.34 204.16 279.41 464.84  765.42  2213 1.00
## y_rep[716]     311.91    2.92 132.56 173.90 287.89  483.75  2058 1.00
## y_rep[717]     302.21    2.80 123.31 172.58 279.42  462.99  1937 1.00
## y_rep[718]     451.41    4.18 184.57 249.24 420.32  686.40  1951 1.00
## y_rep[719]     311.01    2.66 123.06 171.75 293.18  471.58  2142 1.00
## y_rep[720]     205.55    2.03  85.51 114.76 188.87  320.36  1773 1.00
## y_rep[721]     496.65    4.44 193.71 280.65 459.96  758.82  1903 1.00
## y_rep[722]     341.41    3.17 140.86 185.22 317.32  526.77  1977 1.00
## y_rep[723]     228.77    2.20  95.06 126.18 210.27  350.06  1859 1.00
## y_rep[724]     456.62    4.30 195.25 256.43 418.27  700.61  2058 1.00
## y_rep[725]     611.62    6.08 255.27 344.21 563.84  929.14  1765 1.00
## y_rep[726]     201.92    1.90  83.72 111.70 184.86  314.87  1948 1.00
## y_rep[727]     210.01    1.97  89.38 114.92 194.64  322.48  2049 1.00
## y_rep[728]     690.92    6.84 286.20 380.95 638.28 1049.09  1750 1.00
## y_rep[729]     624.20    6.85 278.50 349.16 571.67  944.99  1652 1.00
## y_rep[730]     313.44    2.79 125.55 173.85 295.98  490.60  2030 1.00
## y_rep[731]     232.72    2.04  96.48 126.90 213.62  363.96  2229 1.00
## y_rep[732]     303.73    2.76 125.75 169.09 281.26  464.78  2073 1.00
## y_rep[733]     455.05    4.17 186.66 253.75 418.99  693.99  2004 1.00
## y_rep[734]     438.76    4.17 184.33 247.22 403.26  670.33  1956 1.00
## y_rep[735]     299.69    2.95 126.45 164.46 278.48  458.71  1834 1.00
## y_rep[736]     198.95    1.81  82.34 109.76 183.40  313.27  2071 1.00
## y_rep[737]     209.56    1.99  86.97 115.81 193.26  323.27  1911 1.00
## y_rep[738]     639.01    6.55 284.86 347.82 586.49  986.65  1893 1.00
## y_rep[739]     593.58    5.48 243.75 325.53 549.34  921.99  1978 1.00
## y_rep[740]     207.45    1.91  86.71 115.30 191.44  324.43  2064 1.00
## y_rep[741]     201.50    1.87  82.82 109.04 188.91  309.67  1969 1.00
## y_rep[742]     207.54    2.10  90.64 113.08 190.65  317.19  1857 1.00
## y_rep[743]     452.19    4.58 186.22 253.34 415.69  695.97  1655 1.00
## y_rep[744]     502.86    4.66 204.31 277.99 462.97  761.93  1919 1.00
## y_rep[745]     345.38    3.11 139.31 190.53 319.10  526.23  2013 1.00
## y_rep[746]     300.27    2.82 122.39 165.69 280.70  462.23  1885 1.00
## y_rep[747]     628.10    5.96 262.95 350.97 581.74  961.11  1946 1.00
## y_rep[748]     229.26    2.22  95.85 128.49 210.81  355.18  1860 1.00
## y_rep[749]     309.61    2.79 128.78 168.38 286.50  486.47  2129 1.00
## y_rep[750]     619.82    5.65 256.89 340.48 576.76  934.86  2069 1.00
## y_rep[751]     313.44    2.77 126.16 176.06 294.27  476.93  2072 1.00
## y_rep[752]     454.56    4.28 191.34 250.02 416.01  705.38  2002 1.00
## y_rep[753]     207.86    1.91  83.65 118.15 192.92  320.27  1911 1.00
## y_rep[754]     699.93    6.70 294.15 380.56 646.78 1078.34  1927 1.00
## y_rep[755]     232.22    2.30  94.15 132.30 215.87  351.98  1675 1.00
## y_rep[756]     643.47    6.21 280.55 350.93 589.93  997.00  2040 1.00
## y_rep[757]     346.89    3.47 147.12 187.59 318.57  542.02  1802 1.00
## y_rep[758]     300.79    2.96 126.97 167.66 277.76  461.98  1845 1.00
## y_rep[759]     308.69    3.09 123.75 178.25 287.98  460.61  1601 1.00
## y_rep[760]     442.21    4.54 187.80 243.60 403.00  685.72  1710 1.00
## y_rep[761]     198.36    1.87  81.08 110.57 184.28  306.82  1876 1.00
## y_rep[762]     602.08    5.84 255.71 330.47 554.69  942.17  1919 1.00
## y_rep[763]     443.74    4.00 180.29 249.89 409.54  687.49  2032 1.00
## y_rep[764]     693.40    6.55 285.34 387.80 643.33 1061.02  1899 1.00
## y_rep[765]     616.33    5.77 255.23 348.04 567.80  948.01  1957 1.00
## y_rep[766]     309.42    2.88 125.43 174.59 285.72  477.27  1894 1.00
## y_rep[767]     490.84    4.69 206.91 271.63 449.86  750.47  1949 1.00
## y_rep[768]     206.50    1.91  82.77 115.93 190.65  314.87  1870 1.00
## y_rep[769]     204.86    2.04  85.95 115.50 186.33  314.23  1782 1.00
## y_rep[770]     451.34    4.18 183.26 250.71 419.73  691.14  1920 1.00
## y_rep[771]     231.01    2.22  99.22 125.19 211.10  360.35  1991 1.00
## y_rep[772]     458.04    4.42 193.83 250.35 427.47  701.92  1922 1.00
## y_rep[773]     205.48    2.01  85.28 116.18 189.47  315.36  1804 1.00
## y_rep[774]     304.71    2.70 123.87 169.56 281.45  467.37  2109 1.00
## y_rep[775]     635.88    6.06 268.35 343.98 585.48  981.53  1963 1.00
## y_rep[776]     456.58    4.12 184.76 259.83 420.27  695.44  2010 1.00
## y_rep[777]     350.08    3.12 142.55 193.63 325.31  546.47  2082 1.00
## y_rep[778]     202.12    1.95  84.74 111.63 186.59  313.03  1891 1.00
## y_rep[779]     207.80    1.94  86.23 115.53 191.60  319.97  1970 1.00
## y_rep[780]     292.24    2.74 121.46 164.23 268.07  448.52  1969 1.00
## y_rep[781]     625.68    5.70 255.07 345.12 581.45  946.12  2002 1.00
## y_rep[782]     312.01    3.10 134.35 172.77 283.38  482.80  1877 1.00
## y_rep[783]     204.16    1.95  86.12 113.56 187.54  317.57  1955 1.00
## y_rep[784]     232.81    2.19  98.62 129.49 214.96  356.44  2019 1.00
## y_rep[785]     307.16    2.70 127.48 170.44 283.23  464.76  2222 1.00
## y_rep[786]     209.48    1.94  87.40 115.58 194.03  321.09  2033 1.00
## y_rep[787]     613.36    5.83 248.83 343.20 568.74  938.10  1820 1.00
## y_rep[788]     687.91    6.19 286.62 384.95 634.18 1055.25  2146 1.00
## y_rep[789]     296.88    3.12 127.61 158.97 273.78  458.61  1670 1.00
## y_rep[790]     206.29    1.84  80.88 116.12 192.43  311.50  1937 1.00
## y_rep[791]     448.12    4.39 188.33 247.06 415.78  683.83  1844 1.00
## y_rep[792]     441.12    4.22 186.23 240.86 400.98  693.92  1949 1.00
## y_rep[793]     490.56    4.75 198.48 267.09 458.28  756.27  1744 1.00
## y_rep[794]     341.92    3.25 144.36 191.48 315.11  529.25  1969 1.00
## y_rep[795]     230.45    2.10  95.03 127.46 214.77  352.66  2054 1.00
## y_rep[796]     498.31    5.00 206.56 278.42 465.22  765.30  1703 1.00
## y_rep[797]     309.64    2.87 129.45 171.97 288.36  463.31  2031 1.00
## y_rep[798]     620.67    5.80 257.98 339.30 571.06  981.05  1980 1.00
## y_rep[799]     213.25    1.84  87.48 117.09 196.09  331.06  2261 1.00
## y_rep[800]     460.36    4.28 187.16 257.68 428.85  708.95  1915 1.00
## y_rep[801]     206.36    1.89  84.87 113.69 192.16  313.08  2010 1.00
## y_rep[802]     441.29    3.89 173.55 246.34 414.45  677.00  1989 1.00
## y_rep[803]     600.17    5.60 245.30 331.48 560.76  933.03  1918 1.00
## y_rep[804]     617.02    5.78 259.91 348.09 568.34  958.33  2024 1.00
## y_rep[805]     341.69    2.94 137.55 190.83 318.91  516.61  2188 1.00
## y_rep[806]     440.85    4.08 183.43 241.41 408.58  683.53  2025 1.00
## y_rep[807]     197.45    1.89  83.55 106.92 181.28  306.60  1961 1.00
## y_rep[808]     308.05    2.82 125.36 172.60 285.69  473.27  1970 1.00
## y_rep[809]     301.39    2.76 125.10 169.64 277.45  453.73  2048 1.00
## y_rep[810]     294.20    2.72 124.03 160.38 271.06  460.09  2072 1.00
## y_rep[811]     433.85    3.96 181.29 239.34 398.43  660.59  2096 1.00
## y_rep[812]     308.24    2.93 130.02 172.87 281.17  480.09  1963 1.00
## y_rep[813]     616.59    5.81 257.40 342.02 571.91  952.24  1965 1.00
## y_rep[814]     633.57    5.95 272.77 348.14 577.57  985.03  2104 1.00
## y_rep[815]     202.58    1.99  86.20 112.43 186.04  310.77  1880 1.00
## y_rep[816]     703.28    6.67 300.11 389.85 648.03 1090.23  2024 1.00
## y_rep[817]     230.13    2.15  93.83 126.30 214.03  353.00  1903 1.00
## y_rep[818]     450.21    4.22 187.91 253.58 419.81  685.31  1979 1.00
## y_rep[819]     499.18    4.81 199.96 278.20 463.92  766.24  1729 1.00
## y_rep[820]     598.60    5.75 250.82 335.69 545.79  917.91  1900 1.00
## y_rep[821]     207.24    1.97  86.35 115.85 189.92  315.28  1920 1.00
## y_rep[822]     460.29    4.59 192.67 243.41 429.56  704.60  1759 1.00
## y_rep[823]     204.79    1.93  84.09 112.92 188.21  319.56  1893 1.00
## y_rep[824]     307.86    2.82 128.38 172.86 281.58  482.26  2072 1.00
## y_rep[825]     348.67    3.24 148.57 191.42 319.34  534.22  2101 1.00
## y_rep[826]     696.09    6.63 294.03 380.63 644.49 1052.22  1967 1.00
## y_rep[827]     234.55    2.14  95.07 131.41 217.55  358.39  1970 1.00
## y_rep[828]     309.62    2.81 125.18 174.78 287.23  470.61  1985 1.00
## y_rep[829]     448.66    4.20 190.43 242.96 414.32  687.79  2056 1.00
## y_rep[830]     298.91    2.73 122.87 165.36 277.54  464.56  2032 1.00
## y_rep[831]     459.78    4.43 193.66 257.37 424.02  704.31  1908 1.00
## y_rep[832]     501.43    5.06 208.57 277.81 460.61  774.28  1697 1.00
## y_rep[833]     209.72    1.99  86.54 114.37 193.95  325.32  1883 1.00
## y_rep[834]     304.11    2.77 128.62 170.44 277.66  475.94  2158 1.00
## y_rep[835]     336.15    3.07 138.07 188.27 309.97  516.41  2019 1.00
## y_rep[836]     200.27    1.90  83.16 110.88 185.06  314.96  1911 1.00
## y_rep[837]     623.52    5.68 248.26 348.12 575.60  963.92  1913 1.00
## y_rep[838]     622.16    6.29 263.73 344.08 571.61  964.81  1761 1.00
## y_rep[839]     209.27    1.91  86.17 119.67 194.42  316.56  2033 1.00
## y_rep[840]     314.21    3.03 132.89 170.77 288.53  492.52  1929 1.00
## y_rep[841]     230.33    2.33  97.01 126.42 212.34  358.00  1734 1.00
## y_rep[842]     612.02    5.77 247.27 341.73 568.90  935.21  1834 1.00
## y_rep[843]     626.87    5.84 261.05 348.38 582.50  952.20  1997 1.00
## y_rep[844]     342.35    3.16 139.96 193.38 316.62  520.06  1967 1.00
## y_rep[845]     460.47    4.71 194.18 253.27 423.22  709.78  1698 1.00
## y_rep[846]     600.78    5.73 255.22 332.74 553.69  908.61  1984 1.00
## y_rep[847]     202.95    1.96  85.24 112.31 186.36  313.00  1887 1.00
## y_rep[848]     434.96    3.83 173.35 246.01 405.18  664.56  2050 1.00
## y_rep[849]     702.09    6.87 300.68 393.22 639.04 1103.80  1916 1.00
## y_rep[850]     313.28    2.81 128.45 172.37 290.31  475.66  2092 1.00
## y_rep[851]     451.91    4.57 197.93 244.75 412.92  710.73  1879 1.00
## y_rep[852]     207.79    2.05  87.90 115.64 190.78  318.38  1847 1.00
## y_rep[853]     502.79    4.79 209.54 279.59 462.50  777.22  1917 1.00
## y_rep[854]     299.14    2.93 125.50 161.49 278.13  458.93  1835 1.00
## y_rep[855]     207.56    1.90  86.72 116.27 191.57  317.65  2077 1.00
## y_rep[856]     300.85    2.79 125.66 163.54 279.56  460.76  2028 1.00
## y_rep[857]     208.68    1.93  86.55 117.60 192.89  316.25  2008 1.00
## y_rep[858]     505.28    4.78 208.17 275.83 473.16  774.72  1900 1.00
## y_rep[859]     440.68    4.14 189.09 245.12 402.39  685.40  2084 1.00
## y_rep[860]     207.83    1.92  86.46 116.11 191.43  317.22  2036 1.00
## y_rep[861]     201.71    1.98  85.48 111.86 185.82  314.79  1869 1.00
## y_rep[862]     343.54    3.22 140.77 194.57 316.24  528.99  1908 1.00
## y_rep[863]     596.83    5.31 241.22 330.09 558.26  912.48  2067 1.00
## y_rep[864]     688.27    6.56 290.18 378.02 634.26 1055.25  1955 1.00
## y_rep[865]     235.24    2.29  98.12 129.10 217.63  363.25  1832 1.00
## y_rep[866]     511.48    5.01 214.24 286.70 470.26  783.57  1828 1.00
## y_rep[867]     619.86    6.10 264.75 342.49 569.87  956.09  1881 1.00
## y_rep[868]     212.70    2.04  89.26 116.57 197.61  328.36  1910 1.00
## y_rep[869]     310.08    3.09 135.94 168.62 284.87  479.26  1935 1.00
## y_rep[870]     632.38    6.36 258.83 348.02 587.37  961.16  1655 1.00
## y_rep[871]     690.78    6.88 291.12 375.13 639.15 1059.15  1790 1.00
## y_rep[872]     457.55    4.39 190.98 249.60 426.54  697.82  1893 1.00
## y_rep[873]     300.76    2.86 123.23 165.22 279.10  457.68  1860 1.00
## y_rep[874]     235.85    2.40  97.79 129.00 219.74  366.47  1657 1.00
## y_rep[875]     452.12    4.26 187.44 250.07 417.39  699.41  1937 1.00
## y_rep[876]     345.05    3.13 142.13 193.37 315.98  536.55  2057 1.00
## y_rep[877]     205.60    1.97  84.79 116.65 190.59  314.35  1860 1.00
## y_rep[878]     309.36    2.81 125.85 173.32 286.74  478.79  2003 1.00
## y_rep[879]     201.22    2.22  85.12 110.25 183.34  315.29  1468 1.00
## lp__           354.90    4.70   9.69 345.46 351.93  372.37     4 2.00
## 
## Samples were drawn using NUTS(diag_e) at Mon Apr  6 17:31:05 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
y_rep <- as.matrix(fit.mod3_4, pars = "y_rep")
ppc_dens_overlay(y =data$height,y_rep[1:50, ]) + theme_bw() + theme(legend.text=element_text(size=25), 
                                                                 legend.title=element_text(size=18),
                                                                 axis.text = element_text(size=18),
                                                                 legend.position = c(0.8,0.6))

Comment: \(\alpha\) has very different values between McElreath and Sorensen models..